I think the answer to your question is "yes". Toen-Vezzosi go over this in Proposition 2.4.15, but here is some version of why.
Away from characteristic zero, there's a sharp difference between being a "free" algebra (meaning, having some kind of universal mapping property) and looking like a polynomial algebra. This is because basically every construction has to be replaced with the derived version in order for them to give sensible answers.
For example, in characteristic zero you might take a vector space $V$ over $\mathbb{Q}$ and form the "free" algebra
$$
Sym(V) = \bigoplus_{n \geq 0} Sym^k(V)
$$
and this has the property that maps of commutative ring objects $Sym(V) \to A$ are the same as maps $V \to A$ of underlying objects. This still works even if $V$ is a chain complex.
However, the symmetric power functors aren't well-behaved integrally: there are maps of chain complexes $C \to D$ which are weak equivalences such that $Sym^k(C) \to Sym^k(D)$ isn't a weak equivalence. (These examples aren't particularly hard to find, either.) This means that we have to take the symmetric power
$$
Sym^k(V) = (V^{\otimes k})_{\Sigma_k}
$$
and replace it with the derived version: derived tensor product over whatever your base is, and derived functors of $\Sigma_k$-coinvariants.
Toen-Vezzosi mention the example which they call $\mathbb{F}_p[T]$, which is free on a 1-dimensional vector space over $\mathbb{F}_p$. Here, the derived functors of tensor product don't intrude, but the derived functors of coinvariants do, and they contribute a large amount (namely, the homology of the symmetric groups). In degree zero you're just getting a polynomial algebra on a single generator, but there is extra stuff in positive (homological) degree coming from the Dyer-Lashof operations.
EDIT: Sorry, this object is the "smooth" object.
The thing that looks like a "polynomial" object is the monoid algebra $\mathbb F_p[\mathbb N]$. On homology groups this looks like a polynomial algebra, but it doesn't have simple mapping properties that connect with the notion of smoothness. In particular, they give a description of the derived cotangent complex (which is some measure of how hard it is to build this object out of "free" objects) and it's rather large (certainly not concentrated in degree 0).
Here is an example of a nonconnective $E_\infty$ ring spectrum which, I think, illustrates a key problem. (A more extensive discussion of this phenomenon occurs in Lurie's DAG VIII and in a paper by Bhatt and Halpern-Leinster.)
Let $R$ be an ordinary commutative ring, viewed as an $E_\infty$ ring concentrated in degree zero, and $A$ be the homotopy pullback / derived pullback in the following diagram of $E_\infty$ rings:
Then $\pi_0 A = R[x,y]$ and $\pi_{-1} A$ is the local cohomology group $R[x,y] / (x^\infty, y^\infty)$; all the other homotopy groups of $A$ are zero. As a result, there's a map $R[x,y] \to A$ of $E_\infty$ rings, and any $A$-module becomes an $R[x,y]$-module by restriction.
Here's a theorem. The forgetful map from the derived category $D(A)$ to the derived category $D(R[x,y])$ is fully faithful, and its essential image consists of modules supported away from the origin. This extends to an equivalence of $\infty$-categories.
We could think of this in the following way. The ring $A$ is the $E_\infty$ ring of sections $\Gamma(\Bbb A^2 \setminus \{0\}, \mathcal{O}_{\Bbb A^2})$ on the complement of the origin in affine 2-space over $R$, and the above tells us we actually have an equivalence between $A$-modules and (complexes of) quasicoherent sheaves on $\Bbb A^2 \setminus \{0\}$.
Here are some takeaways from this.
Nonconnective ring spectra are actually quite natural. Global section objects $\Gamma(X, \mathcal{O}_X)$ are usually nonconnective, and we're certainly interested in those.
The above says that even though the punctured plane is not affine but merely quasi-affine, it becomes affine in nonconnective DAG. This is a general phenomenon.
Solely on the level of coefficient rings, the map $R[x,y] \to A$ looks terrible. It is indistinguishable from a square-zero extension $R[x,y] \oplus R[x,y]/(x^\infty,y^\infty)[-1]$. (There is more structure that does distinguish them.)
Many of the definitions as given in DAG for a map are given in terms of the effect (locally) of a map $B \to A$ of ring spectra (e.g. flatness, étaleness, etc etc). For connective objects, this works very well. However, we have just shown that for nonconnective objects, a map of ring spectra may have nice properties—the map $Spec(A) \to Spec(R[x,y])$ should be an open immersion!—which are completely invisible on the level of coefficient rings. This goes for the rings themselves and doubly so for their module categories.
If I have one point here, it is that trying to give definitions in nonconnective DAG in terms of coefficient rings is like trying to define properties of a map of schemes $X \to Y$ in terms of the global section rings $\Gamma(Y,\mathcal{O}_Y) \to \Gamma(X,\mathcal{O}_X)$. This makes nonconnective DAG fundamentally harder.
So far as your question A, this places me somewhere in between your two options (1) and (2). I don't think (1) is right because I think that nonconnective objects are much too important; I have a mild objection to the language in (2) because I don't think that nonconnective objects are straightforward.
Best Answer
I'm not quite sure what kind of answer you're expecting, but here is a geometric example that may help to grasp some intuition.
In differential geometry, when an intersection is badly behaved (e.g. it doesn't have the expected dimension) one can geometrically perturbe one of the two factors. For instance, if you are intersecting tow submanifolds $X,Y\subset Z$, and if $X$ is locally given as the zero of some functions $X\overset{\text{loc}}{=}\{f_1=\dotsb=f_k=0\}$, you may want to introduce a deformation $X_{t_1,\dotsc,t_k}$ of $X$ defined as $$ X_{t_1,\dotsc,t_k}\overset{\text{loc}}{=}\{f_1=t_1,\dotsc,f_k=t_k\}. $$ One of the main idea of derived geometry is to replace these deformation/perturbation parameters $t_i$'s by a homological perturbation: $$ X_{t_1,\dotsc,t_k}\overset{\text{loc}}{=}\{f_1=d\tau_1,\dotsc,f_k=d\tau_k\}, $$ with $\operatorname{deg}(\tau_i)=-1$ (my degree convention is cohomological).
Homological perturbation has two advantages above geometric perturbations:
it can be made functorial.
it exists even in the (quite non-flexible) algebraic setting, where geometric perturbation may not exist.
Let's try to apply informally the above reasoning to the case discussed in Jon Pridham's comment: consider $X=Y=\{x=0\}$ inside $Z=\mathbb{A}^1=\operatorname{Spec}(k[x])$. You deform $\{x=0\}$ to $\{x=d\tau\}$ and then proceed with the intersection of $\{x=d\tau\}$ with $\{x=0\}$, and get $\{d\tau=0\}$, which is just a ($k$-)point (it is $0$ in $\mathbb{A}^1$) together with a self-homotopy (given by $\tau$). This is indeed the "space" of derived loops in the affine line that are based at $0$.
I apologize for self-promoting, but you can read an informal account of how to view derived self-intersections as some kind of based loop spaces in the introduction of Calaque, Căldăraru, and Tu - On the Lie algebroid of a derived self-intersection.