findaphd.com is an excellent website to search for PhD studentships (PhD funding on specific projects) in the UK. Also, UK universities often advertise PhD studentship opportunities in their "job vacancies" section. However, after much googling, I have yet to find similar sites or departmental advertisements for PhD studentships in the USA.
I don't suppose there is a similar site for the USA that I missed during my search?
Are there even off-cycle PhD studentships for specific projects in the USA, like those in the UK?
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
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.