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
Yes, they are more general. This is in fact already the case with ordinary rings. Let's call a classically-ringed $\infty$-topos which is locally the Zariski $\infty$-topos of an affine scheme an $\infty$-scheme. A classical scheme is then the same as a $0$-localic $\infty$-scheme. To construct an $\infty$-scheme which is not classical, let $F$ be any object in the Zariski $\infty$-topos $\mathrm{Shv}(X)$ of a classical scheme $X$. Then the slice $\infty$-topos $\mathrm{Shv}(X)_{/F}$ with the restricted sheaf of rings is an $\infty$-scheme (it is covered by open subschemes of $X$), which is classical iff $F$ is $0$-truncated. This same construction works to define spectral $\infty$-schemes that are not $0$-localic.
A theorem of Lurie (Theorem 2.3.13 in DAG V) implies that every $\infty$-scheme is such a slice over a $1$-localic $\infty$-scheme. I do not know if there exist $1$-localic examples that are not slices over classical schemes. If one works with the étale topology, however, this theorem implies that every DM $\infty$-stack is a slice over a classical DM stack.
$\infty$-schemes can also be identified with the full subcategory of $\mathrm{Shv}_{\mathrm{Zar}}(\mathrm{CRing}^\mathrm{op})$ consisting of those Zariski sheaves (of spaces) $F$ that admit an effective epimorphism from a coproduct of representable sheaves $R_i$ such that each $R_i\to F$ is "representable by slice $\infty$-topoi" (see Proposition 2.4.17(6) in loc. cit.). Unlike with classical schemes, however, such sheaves do not usually satisfy descent with respect to the étale topology, because higher Zariski and étale cohomology generally disagree.
ETA: This last remark implies that the natural functor from $\infty$-schemes to DM $\infty$-stacks is not fully faithful. If it were, then the functor of points of any $\infty$-scheme would be the same as the functor of points of the associated DM $\infty$-stack, which is an étale sheaf.