- Are websites (if they do something which hasn't been done before)
eligible for a patent?
Only if it meets the requirements for patentability such as originality, non-obviousness, non-publication, and utility. The fact that a website merely does something that hasn't been done before, however, isn't enough to make an idea patentable.
Also, generally, it is the idea behind a website that is patentable and not the website itself. A website will usually have copyright protection, but will only rarely be eligible for patent protection.
The very act of using the website in a manner accessible to the public which may publicly disclose the idea that you seek to patent threatens its patentability.
- If not (1), then can we patent the idea which gave birth to the
If the idea behind the website is patentable, yes.
- Does the use of the university's servers grant the ownership of the
patent to them?
It is conceivable that there could be a separate express agreement between you and the university that some of the work you do as part of the university program at its direction could be considered "work for hire" that belongs to the university (e.g. if it arose from your work as a research assistant and there is an express RA contract that says so).
But, in general, the mere use of the university's services does not give it any rights in a patent that you develop using those servers.