How to search for USA PhD Studentships?
There is no such site because it is unnecessary. In Europe, a Ph.D. student is like an employee and is admitted to work on a specific project. In the USA, a Ph.D. student is admitted in a manner similar to an undergraduate (based on general considerations) and is not necessarily attached to a particular project or even a particular advisor.
Hence, you could say that every US department has a number of studentships available each year, always at the same time (start of fall semester) and you apply for all of them simply by submitting an admission application to the program.
Thus the answers to your numbered questions are
- No.
- No.
Disclaimer: there are probably exceptions, but what I've written applies 99% of the time.
One additional note: I disagree with the comment below that claims that individual faculty do not decide which students are admitted. In every program I know, a small committee of program faculty makes all the decisions. If some faculty member (whether on the committee or not) really wants a particular applicant, that applicant will be admitted. My own admission came on the same day that I contacted a faculty member (and as a direct result). Of course, you should always follow this good advice when contacting faculty you'd like to work with.
The closest that you can come to this in the US is listings of graduate fellowships, which is more closely related to the issue of funding rather than admissions. Pretty much every department has an annual cycle for admitting students, as David mentions.
However, funding models vary widely, so acquisition of an external fellowship can make a significant difference in the kinds of projects one can take (since positions tend to be closely tied to specific projects, as a result of the grant model in the US).