Where do sublinear time algorithms fit in the picture of complexity theory?
Sublinear time algorithms (as well as sublinear space algorithms) are indeed an active area of research. The field of property testing covers much of the sublinear time regime, and I'll point you to the property testing blog to learn about recent work in the area. 

The question about complexity theory relating to this is very interesting, and is also an active area of research. P vs NP might not exactly be the right analogy here, but you're right that the boundary between computation and verification is something where sublinearity changes things. 

In particular, you can look at a PCP as "kind of" doing something sublinear, in that it only inspects a few bits of a long proof in order to check the prover's claim. More generally, there's been recent work prover-verifier systems where the verifier runs in sublinear time. Some references that are worth perusing:


Interactive proofs of proximity: delegating computation in sublinear time
Non-Interactive Proofs of Proximity