I don't know what Toën was talking about, but I suspect that it was about finiteness conditions for Artin stacks: the problem is that the usual finiteness conditions we look at for schemes (like the notion of constructibility for l-adic sheaves) do not extend to stacks in a straightforward way, which gives some trouble if one wants to count points
(i.e. to define things like Euler characteristics). Some notions of finiteness are developed to define Grothendieck rings of Artin stacks (e.g. in Toën's paper arXiv:0509098 and in Ekedahl's paper arXiv:0903.3143), which can be realized by our favourite cohomologies (l-adic, Hodge, etc), but the link with a good notion of finiteness for categories of coefficients over Artin stacks (l-adic sheaves, variation of mixed Hodge structures) does not seem to be fully understood yet, at least conceptually (and by myself).
As for finiteness conditions for sheaves (in some homotopical context), the kind of properties we might want to look at are of the following shape.
Consider a variety of you favourite kind X, and a derived category D(X) of sheaves over
some site S associated to X (e.g. open, or étale, or smooth subvarieties over X etc.).
For instance, D(X) might be the homotopy category of the model category of simplicial sheaves, or the derived category of sheaves of R-modules.
Important finiteness properties can be expressed by saying that for any U in the site S,
we have
(1) hocolimᵢ RΓ(U,Fᵢ)= RΓ(U,hocolimᵢ Fᵢ)
where {Fᵢ} is a filtered diagram of coefficients. If you are in such a context, then
you can look at the compact objects in D(X), i.e. the objects A of D(X) such that
(2) hocolimᵢ RHom(A,Fᵢ)= RHom(A,hocolimᵢ Fᵢ)
for any filtered diagram {Fᵢ}. In good situations, condition (1) will imply that the category of compact objects will coincide with constructible objects (i.e. the smallest subcategory of D(X) stable under finite homotopy colimits (finite meaning: indexed by finite posets) which contains the representable objects).
Sufficient conditions to get (1) are the following:
a) For simplicial sheaves (as well as sheaves of spectra or R-modules...), a sufficient condition is that the topology on S is defined by a cd-structure in the sense of Voevodsky (see arXiv:0805.4578). These include the Zariski topology, the Nisnevich topology, as well as the cdh topology (the latter being generated by Nisnevich coverings as well as by blow-ups in a suitable sense), at least if we work with noetherian schemes of finite dimension. Note also that topologies associated to cd structures define what Morel and Voevodsky call a site of finite type (in the language of Lurie, this means that, for such sites, the notion of descent is the same as the notion of hyperdescent: descent for infinity-stacks over S can be tested only using truncated hypercoverings; this is the issue discussed by David Ben Zvi above).
In practice, the existence of a cd structure allows you to express (hyper)descent using only Mayer-Vietoris-like long exact sequences (the case of Zariski topology was discovered in the 70's by Brown and Gersten, and they used it to prove Zariski descent for algebraic K-theory).
b) For complexes of sheaves of R-modules, a sufficient set of conditions are
i) the site S is coherent (in the sense of SGA4).
ii) any object of the site S is of finite cohomological dimension (with coefficients in R).
The idea to prove (1) under assumption b) is that one proves it first when all the Fᵢ's are concentrated in degree 0 (this is done in SGA4 under assumption b)i)).
This implies the result when the Fᵢ's are uniformly bounded. Then, one uses the fact, that, under condition b)ii), the Leray spectral sequence converges strongly, even for unbounded complexes (this done at the begining of the paper of Suslin and Voevodsky "Bloch-Kato conjecture and motivic cohomology with finite coefficients").
This works for instance for étale sheaves of R-modules, where R=Z/n, with n prime to the residual characteristics. Note moreover that, in the derived category of R-modules, the compact objects (i.e. the complexes A satisfying (2)) are precisely the perfect complexes.
The fact that the six Grothendieck operations preserves constructibility can then be translated into the finiteness of cohomology groups (note however that the notion of constructiblity is more complex then this in general: if we work with l-adic sheaves
(with Ekedahl's construction, for instance), then the notion of constructiblity does not agree with compactness anymore). However, condition (1) is preserved after taking the Verdier quotient of D(X) by any thick subcategory T obtained as the smallest thick subcategory which is closed under small sums and which contains a given small set of compact objects of D(X) (this is Thomason's theorem). This is how such nice properties survive in the context of homotopy theory of schemes for instance.
Note also that, in a stable (triangulated) context, condition (2) for A implies that we have the same property, but without requiring the diagrams {Fᵢ} to be filtering.
For your second question, the extension of a cohomology theory to simplicial varieties is automatic (whenever the cohomology is given by a complex of presheaves), at least if we have enough room to take homotopy limits, which is usually the case (and not difficult to force if necessary). The only trouble is that you might lose the finiteness conditions, unless you prove that your favorite simplicial object A satisfies (2). The fact that Hironaka's resolution of singularities gives the good construction (i.e. gives nice objects for open and/or singular varieties) can be expained by finiteness properties related to descent by blow-ups (i.e. cdh descent), but the arguments needed for this use strongly that we work in a stable context (I don't know any argument like this for simplicial sheaves). The fuzzy idea is that if a cohomology theory satisfies Nisnevich descent and homotopy invariance, then it satisfies cdh descent (there is a nice very general proof of this in Voevodsky's paper arXiv:0805.4576 (thm 4.2, where you will see we need to be able to desuspend)); then, thanks to Hironaka, locally for the cdh topology, any scheme is the complement of a strict normal crossing divisor in a projective and smooth variety. As cdh topology has nice finiteness properties (namely a)), and as
any k-scheme of finite type is coherent in the cdh topos, this explains, roughly, why we get nice extensions of our cohomology theories (as far as you had a good knowledge of smooth and projective varieties). If we work with rational coefficients, the same principle applies for schemes over an excellent noetherian scheme S of dimension lesser or equal to 2, using de Jong's results instead of Hironaka's, and replacing the cdh topology by the h topology (the latter being obtained from the cdh topology by adding finite surjective morphisms): it is then sufficient to have a good control of proper regular S-schemes.
Best Answer
Here are some comments about the use of topologies in motivic homotopy theory. This is based on the discussion in Morel-Voevodsky's "A^1-homotopy theory of schemes" p.94-95 (MV below), I only add some background and references. I also comment on the differences between the development of the unstable and stable theories. I am not an expert, so please comment/edit out innacurate statements.
First some model category generalities. Various model structures on categories of simplicial (pre)sheaves on any Grothendieck site $(\mathcal{C},\tau)$ can be constructed. For a survey see:
http://nlab.mathforge.org/nlab/show/model+structure+on+simplicial+presheaves
and the references therein. Many of those are Quillen equivalent and can be thought as different presentations of the $(\infty,1)$-category of $(\infty,1)$ $\tau$-sheaves on $\mathcal{C}$. As usual in model category theory, they are suited for deriving different functors (i.e. some natural functors will be Quillen with respect to some structures but not others).
The resulting model categories are all left proper, simplicial and combinatorial, so by a theorem of Smith (see http://nlab.mathforge.org/nlab/show/Bousfield+localization+of+model+categories#Existence )) left Bousfield localizations at a set of maps exist. In particular if $\mathcal{C}$ is a small category of schemes over a fixed $S$ which is stable by $S$-fiber products and contains $\mathbb{A}^1_S$, one can localize at the set of maps $X\times_S \mathbb{A}^1_S\rightarrow X$.
The Morel-Voevodsky category $\mathcal{H}(S)$ can be obtained by this procedure with $\mathcal{C}=Sm/S$, $\tau=Nis$ and using as starting point the so-called injective model structure on simplicial sheaves. It should be clear from the above that many variants are possible, some giving alternative models for $\mathcal{H}(S)$ (using projective model structures, simplicial presheaves instead of sheaves, or even more exotic choices like cubical presheaves, etc.) and some giving different categories (using another topology: Zariski, étale, using the category of all schemes instead of smooth ones, etc.).
The passage to the stable theory via model categories of spectra is also formal (in the sense that it can be done in great generality and with variants) but rather subtle. See e.g. Riou "Catégorie homotopique stable d'un site suspendu avec intevalle" or Ayoub's thesis.
The question now becomes: in which respects are $\mathcal{H}(S)$ and $\mathcal{SH}(S)$ nicer than the alternatives ? And how much does this depend on the choice of the Nisnevich topology ?
Here are some possible answers:
1) Characterisation of $\tau$-local simplicial (pre)sheaves. For a general site (and in particular for the étale site over a general scheme), this is a complicated condition which can be expressed only in terms of a descent condition for hypercovers (see Dugger-Hollander-Isaksen, "Hypercovers and simplicial presheaves"). For the Zariski and the Nisnevich site, one can show a "Brown-Gersten" property: $\tau$-locality can be rephrased as some squares of simplicial sets associated to $\tau$-distinguished squares being homotopy cartesian. See MV proposition 1.16 for the case of the Nisnevich topology.
Aside: this argument has been abstracted by Voevodsky in "Homotopy theory of simplicial sheaves in completely decomposable topologies" and used to compare $\mathcal{H}(k)$ for $k$ a field admitting resolution of singularities with a similar category defined with $\mathcal{C}=Sch/k$ and $\tau=cdh$. In the setting of triangulated categories of mixed motives, this argument can be pushed in various directions to exploit various forms of resolution of singularities, see Cisinski-Deglise, "Triangulated categories of mixed motives", section 3.3.
The Brown-Gersten property in turn plays an important role in MV. For instance, it implies that the property of being $\tau$-local is stable by filtered colimits. It is also used in MV to construct an explicit A^1-localisation functor and to study the functoriality of $\mathcal{H}(S)$. On the other hand I am not sure if the use of the Brown-Gersten property for all this is unavoidable: at least in the context of $\mathcal{SH}(S)$, there are arguments which do not use this (see Riou's paper mentioned above or Ayoub's thesis) and which consequently give some results for the étale topology. On the other hand, in following works of Morel on $\mathcal{H}(k)$, there are really substantial applications of the Brown-Gersten condition, see "A^1-algebraic topology over a field" Chap. 8 and App. A.
2) Homotopy purity (or localisation). The proof of this major theorem in MV requires a topology at least as fine as the Nisnevich topology and to work with $\mathcal{C}=Sm$. The idea is that to reduce the theorem to the case of the closed immersion $Z\rightarrow A^n_Z$ (where one can write explicit $A^1$-homotopies), one uses the local structure of smooth pairs in the étale topology (cf EGAIV 17.12.2) and the fact that étale morphisms to Henselian local schemes (i.e. points of the Nisnevich site) which have a section on the closed point have a section.
3) Compactness properties. The cohomological dimension of the small Nisnevich site on a noetherian scheme is bounded by the Krull dimension, see MV 1.8 and Thomason-Trobaugh E.6.c. This is very different from the case of the étale topology. This has important consequences for the stable theory. It implies in particular that $\mathcal{SH}(S)$ is compactly generated. See e.g. the last paragraphs of Ayoub's thesis. I do not know if there are analoguous statements to be made in the unstable case.
4) Nisnevich descent for motivic cohomology and algebraic K-theory. The descent properties of algebraic K-theory have been studied long before motivic homotopy theory. The definitive, pre-$A^1$-homotopy result (Nisnevich descent for algebraic K-theory of regular schemes) is in Thomason-Trobaugh, and this is used as an input in the proof of representability of algebraic K-theory in MV p.139. On the other hand, algebraic K-theory with integral coefficients does not satisfy étale descent and hence cannot be represented in the étale $A^1$-homotopy category.