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.

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.