Domain Controllers and SQL Server 2000 Problem

Hey everyone...I know nothing (and I do mean nothing) about security for SQL Server 2000, so you'll have to pardon me for anything silly I may post. I have an instance of SQL Server 2000 running on a Windows 2003 Server-based Veritas cluster (we'll call the live node A, the redundant node B and the virtual node C). Twice in the past month, we've been getting errors in our jobs that say "SQL Server does not exist or access denied". Both times, within 24 - 30 hours of us noticing these errors, we get "Login failed for user '(null)'. Reason: Not associated with a trusted SQL Server connection". Trying to access the Win2K3 server, our application, or the database (through Query Analyzer) all generate the same message. I have been told that this is a problem with our Domain Controller, which leads me to my questions.
1) I understand that a Domain Controller authenticates the users and jobs accessing the database. But why does it lock out access to the server itself?
2) How do I know if the Domain Controller is on the same server as our database (which is apparently a bad thing)? Where should I have it - can it be on a different drive on the same server, or does it have to be completely removed?
3) Is there something that we can do to permanently fix this or am I going to have to live with these errors?
4) A friend suggested that I need to check if unauthorised access to my database has been attempted - how do I do this, if in fact I need to?
5) What other things do I need to be aware of? I need to have one or two plan(s) of action for my boss by end of day tomorrow, and I don't even know what I'm dealing with here. Really appreciate any and all suggestions / tips / advice / etc.
TIA
Chris