|"I also know that the off site network admin has the capability to see what is being done via the web from each computer in each apartment."|
This is a false conclusion.
The best they could do is review the router logs which may list what domains you visit. It would require a capture machine between you and the router to then review the transmissions. I don't see that happening in your situation.
When you access your banks site its via a secure ssl link. No chance this traffic is being intercepted. You should never store banking/credit information on your hard drive.
Engaging the windows firewall would let you know if anything is trying to access your pc.
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