Hi, so I don't fully understand your question #1. Do you mean an abitrary vector in the full Hilbert space? If so, yes one could do this but there's no such function defined in ITensor currently. You would have to write it (basically it would be a sequence of SVDs).
About #2, there's no way I know of to exponentiate MPO's exactly in general. For the special case of a Hamiltonian where all the terms commute it can be done, though. (See the paper by Pirvu, Murg, Cirac, and Verstraete from 2010 for details.)