Such structures have been investigated at depth (welcome to the club!). Let me try to answer some fo of your questions.
1) First, let me suggest that you look at Bill Arveson's book on this subject, Noncommutative Dynamics and E-semigroups. Arveson is considered the pioneer of modern work on these structures, and you will enjoy his book.
2) An "infinitesemal generator" can mean various things, but in some sense, remarkably, it doesn't always exist. Arveson's book contains some treatment of this issue, known as the existence of type III examples, which are due to R. T. Powers and B. Tsirelson (there is also recet work of M. Izumi).
3) Do we need continuity conditions? Measurability conditions suffice. The condition is that the bundle {E_t}_{t>0} be isomorphic, as a measurable bundle of Hilbert spaces, to the trivial bundle (0,infinity) X H_0, with H_0 some fixed Hilbert space, plus compatibility of the measurable structure with addition, multiplication, etc.
4) Extensive research has been carried out also in the case where the E_t are Hilbert bimodules (C^*-correspondences). Search the ArXiv for works of Michael Skeide, or Paul Muhly and Baruch Solel.
5) I should also mention: these product systems arise naturally in the study of, give rise to, and are in a one-to-one correspondence with semigroups of *-endomorphisms on von Neumann algebras.
