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)?