The answer to all of your questions is yes. This is a theorem announced by Kirillov in
Kirillov, A. A. Representations of the infinite-dimensional unitary group. Dokl. Akad. Nauk. SSSR, 1973, 212, 288-290
and proved by Olshanski in
Olshanski, G. I. Unitary representations of the infinite-dimensional classical groups $U(p,\infty)$, $SO_0(p,\infty )$, $Sp(p,\infty)$, and of the corresponding motion groups. Funktsional. Anal. i Prilozhen., 1978, 12, 32-44, 96
Edit: This answer applies to continuous unitary representations of $U(\mathcal{H})$, where the latter group is equipped with the strong operator topology (which is the usual topology on this group). However, if you are only interested in representations on a separable Hilbert space, the continuity assumption can be dropped, as is shown in arXiv:1109.1200.