networking - Limited connectivity issues in network, need help sorting out the root cause

I have an issue with our house network that I can't figure out how to solve. I'll try to detail the symptoms as best I can, but please feel free to ask me for clarifications and I'll try to find any data you need.

My issue is one of connectivity. I seem to be having trouble connecting to peers and certain servers. One symptom is the fact that torrents can literally take several days to start, but once they're connected to a peer it will stay connected and keep downloading at a fast and steady pace.

The other large symptom is the fact that Xbox Live is incredibly hard to use on my Xbox 360. Connecting to Live takes multiple attempts (sometimes 5, sometimes I give up after trying to connect for 90 minutes straight), as does every other connection step along the way; trying to join a chat party, getting into a multiplayer match, etcetera. For some reason, I'm disconnected every time I go to the dashboard too, so my connection doesn't like whatever it is the Live server is doing whenever you switch games. Additionally, I'm kicked off of Xbox Live every 2 hours. (However, this time limit fluctuates; sometimes I can be on for 6 hours straight with no issues and sometimes I'm thrown out after just 10 minutes. 90% of the time it's 2 hours though.)

A major problem on top of this is the fact that my entire apartment building is on one gigabit connection, routed through a NetGear ProSafe router owned by my landlord. I can't just sign up for my apartment's own connection, as the house is a converted office building and isn't wired with anything but CAT5 cables running from the basement.

One more detail is that this seems to be load-based. In early mornings, when people are still asleep, the issues are almost completely gone. In prime time evenings, though, trying to use the connection is a nightmare.

Is the issue that the ProSafe can't handle about 50 apartments trying to connect, or is there a setting that's too strict and ends up bottlenecking the house, causing these issues? Any tips or clues would be appreciated, as I can neither get my own connection nor move to a different place very easily and this issue is absolutely maddening for a huge gamer like myself.