So one of the major problems with the categories of schemes and algebraic spaces is that the "correct quotients" are oftentimes not schemes or algebraic spaces. The way I've seen this sort of thing rectified is either by moving a step up the categorical ladder or by defining some nonstandard quotient to "fix" things. That is, the (sheafy) quotient of a scheme by an étale equivalence relation is an algebraic space, and the (stacky) quotient of an algebraic space by a smooth groupoid action is an algebraic stack (and it is my understanding that these descriptions characterize alg. spaces and stacks up to equivalence).
Derived algebraic geometry gives us a number of powerful tools and has some very nice features: We have a whole array of new, higher dimensional, affine objects (coming from simplicial commutative rings), and a good supply of higher-categorical objects, which we get "all at once", as it were, rather than piecemeal one-level-at-a-time descriptions.
Does the theory of derived algebraic geometry give us enough "n-categorical headroom" (to quote a recent comment of Jim Borger) to take quotients of geometric objects "with reckless abandon" (not a quote of Jim Borger)?
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There is more than one way that derived algebraic geometry generalizes ordinary algebraic geometry. The new affines don't help you much with quotients, which are (homotopy) colimits, but they give you well-behaved intersections, which are (homotopy) limits. On the other hand, you can consider functors from affines (new or old) to a category like simplicial sets that has better quotient behavior than plain sets. This gives you a notion of derived stacks, and I believe they behave well under many colimits.
I'm not sure what you mean by "reckless abandon". I tend to make mistakes when I'm not careful with my mathematics, even if I'm looking at derived algebraic geometry.