If you define the local university proxy in Firefox, every byte of traffic generated by Firefox will be available to them. And that includes:
Time and date of every packet
Source IP (that's you) and destination of every connection
If not encrypted (SSL/TLS/VPN), every single byte in clear
Amount of data transmitted
If the proxy wants to log everything, it can even log all your session for later examination.
Having the capability to log everything does not mean they do log. That would cost a lot in terms of processing power and storage media, so generally they log the origin and destination addresses, protocol, and timestamps. But that is enough to see if you are accessing a research journal or a recreational magazine.
Generally, only the traffic generated by Firefox would be routed through the proxy. But there are some programs that will try to access the Internet directly, and will search for proxy configuration in case the direct connection fails. If you have programs of this kind, they can reach the proxy by themselves.