Let $M$ be $S^7$ with an exotic smooth structure. Since one can smoothen maps, there exist smooth maps $f:S^7\to M$ which are homotopic to the identity (relative to a base point, if you want).

Can one make explicit one such map? Can such a map be an homeomorphism?

Little addendum. The smooth homeomorphism constructed in Ryan's answer below is of course not a diffeomorphism.

Does one have some control on the non-smooth locus of the inverse of smooth homeomorphims, or on the type of their non-smoothness there?

The inverse of Ryan's map is non-smooth only at the bad pole and I guess the initial map $h$ arises as the "conical differential" of the map there, so the singularity there is pretty bad. Maybe one can find other smooth homeomorphisms whose inverse has a larger non-smooth locus but with tamer non-smoothness there?