One standard answer*, in which any reasonable (characteristic $0$ — I haven't thought about any other case) algebraic category can be given a simplicial structure, is the following.
Let $\mathbb Q[\Delta^k] = \mathbb Q[t_0,\dots,t_k,\partial t_0,\dots,\partial t_k] / \bigl\langle \sum t_i = 1,\ \sum\partial t_i = 0\bigr\rangle$ denote the differential graded commutative algebra (dgca) of polynomial forms on the standard $k$-simplex. Here $t_i$ are in (co)homological degree $0$, and their derivatives $\partial t_i$ are in degree $\pm 1$ depending on whether you prefer homological or cohomological conventions. It is straightforward to check that $\mathbb Q[\Delta^k]$ has (co)homology only in degree $0$, where it is $1$-dimensional. Moreover, there are natural face and degeneracy maps between different $\mathbb Q[\Delta^k]$, making $\mathbb Q[\Delta^\bullet]$ into a simplicial dgca.
Given two $L_\infty$ algebras $V,W$ (or, really, objects of any reasonable category of "algebras"), one then defines the space of maps $V \to W$ to be the simplicial set
$$ \hom_\bullet(V,W) = \hom(V,W[\Delta^\bullet]),$$
where $W[\Delta^\bullet] = W\otimes_{\mathbb Q} \mathbb Q[\Delta^\bullet]$ is the $L_\infty$ algebra $W$ base-changed to live over the $k$-simplex. It is reasonably straightforward to prove that this simplicial set satisfies the Kan horn-filling condition, at least when $V$ is "quasifree" — in particular, in your situation of "nonlinear $L_\infty$-algebra homomorphisms", the Kan condition is always satisfied.
Before I spell this out, I'm going to change your notation. What you called $f_k$ I will call $f^{(k)}$, since it plays the role of the "$k$th Taylor coefficient of $f$". That way, I can ask "what is a homotopy between two morphisms $f_0,f_1 : V \to W$ of $L_\infty$-algebras?"
The answer is the following data: (1) a (nonlinear) homomorphism $f_t: V \to W$ that depends polynomially on a parameter $t$, with the correct evaluations $f_t|_{t=0} = f_0$ and $f_t|_{t=1} = f_1$; (2) maps $\phi^{(k)}_t : V \to W[1]$ (or maybe I mean $[-1]$), also depending polynomially on the parameter $t$. These data must satisfy a certain ODE of the form:
$$ \frac{\mathrm d}{\mathrm d t} f_t = \operatorname{ad}_{f_t}(\phi_t) $$
Of course, this is really an infinite sequence of equations (which are equations to things that depend polynomially on $t$). The $k$th entry on the left hand side is $ \frac{\mathrm d}{\mathrm d t} f_t^{(k)}$. On the right hand side, the $k$th entry is computed as follows (up to a sign which I don't feel like working out). Consider the equations saying that $f_t$ is a homomorphism; one of these equations is an equation of things with $k$ inputs $x$. Sum over all ways to replace, in each summand in this equation, one of the occurrences of an $f$ by a $\phi$. Such a sum is what I mean by the right-hand side. In short-hand, what I mean is: there is (a sequence of) equations $M(f)$, such that $f$ is a homomorphism iff $M(f) = 0$. The right hand side is $\frac{\partial M}{\partial f} \cdot \phi$.
In good situations like yours, all the ODEs that occur when studying $\hom_\bullet(V,W)$ are pretty well behaved. In particular, their integral forms are contraction mappings in the appropriate sense, so the initial and boundary value problems are pretty easy to analyze formally.
*Here is an important (elementary) exercise to work out if you want to understand this "standard answer." Consider just the category of chain complexes. Then, for $k \geq 0$, $\pi_k\bigl( \hom(V,W[\Delta^\bullet])$ is the space of chain maps $V \to W[\pm k]$ modulo chain homotopies, i.e. it is $\mathrm{H}_k(\underline\hom(V,W))$, where $\underline\hom$ denotes the chain complex of all linear maps $f: V \to W$ with differential $f \mapsto [\partial,f] = \partial_W\circ f -(-1)^{\deg f}f\circ \partial_V$. (Whether the shift should be $[k]$ or $[-k]$, and whether I mean $\mathrm H_{\pm k}$ or $\mathrm H^{\pm k}$ or ..., depend on your conventions, so I didn't work them out.)
Best Answer
Q1: The conventional theory of homotopy algebras is built on the premise that the lower-degree operations dominate over the higher-degree ones, in some sense. A discussion of this can be found in the introduction to my preprint "Weakly curved $\mathrm A_\infty$-algebras over a topological local ring", http://arxiv.org/abs/1202.2697. This does not answer your question fully, but explains the underlying ideology to some extent.
Q2: The theory of curved DG-algebras/curved DG-coalgebras and the co/contraderived categories of CDG-modules/comodules/contramodules over them is built on the premise that the (co)multiplication dominates over the differential (and the differential dominates over the curvature). So the higher-degree operations dominate over the lower-degree ones in these "theories of the second kind". On CDG-coalgebras or DG-coalgebras, there is even a model structure with such weak equivalences (though a precise definition is more complicated and maybe does not accord to what you describe in the question). See my memoir "Two kinds of derived categories, ...", http://arxiv.org/abs/0905.2621.
Q3: It is indeed true that the homotopy category of DG Lie algebras is equivalent to the homotopy category of $\mathrm L_\infty$-algebras, though perhaps for reasons more complicated than described in the question. Similarly, the homotopy category of associative DG-algebras is equivalent to the homotopy category of $\mathrm A_\infty$-algebras.
Basically, with any $\mathrm A_\infty$-algebra one can naturally associate a much bigger DG-algebra quasi-isomorphic to it; while for any DG-algebra one can, if one wishes, construct a (generally speaking) much smaller $\mathrm A_\infty$-algebra quasi-isomorphic to it (in addition to the obvious option of viewing a DG-algebra as an $\mathrm A_\infty$-algebra with vanishing higher operations).
But of course, one cannot just forget the higher operations of an $\mathrm A_\infty$-algebra and obtain a quasi-isomorphic DG-algebra, firstly because the multiplication in an $\mathrm A_\infty$-algebra need not be associative, and secondly, even if it is, the identity map cannot be extended to an $\mathrm A_\infty$-morphism (in most cases).