In 2D and 3D quibit models, string-net condensation can happen. In 3D or higher models, is it possible for surfaces (instead of just strings) to condense?
Yes it seems that it is possible. See for example arXiv:cond-mat/0411752 where some models are constructed, but I (naively) think one can readily generalize the whole logic of Levin-Wen models to any dimension and any types of branes (the mathematical input might then be something more exotic than tensor categories?). But I think that excitations of these models will generically be extended objects, for example the boundaries of open membranes will be string-like object. One could also imagine that low-energy effective theories if these models might not be conventional (topological) field theories, but string- and brane-field theories (this is however not the case in the above reference it seems).
I don't know enough about this to say much more, but I know others on this site know a lot more. I hope they will give their take on this question.