Doesn't sound stupid, every console comes with a 3 year cover for the RRoD so don't worry it's free.
The Xbox Live sign in problem is a problem with the service itself and not your 360. Since the scheduled maintainence that was preparing for the November release of the NXE update on Monday, there has been a fair number of unique users having this same problem.
However, if you had RRoD once, it will happen again in the near future so expect it to fail within the next few weeks. If it doesn't fail, check said Xbox for rabbits feet/four leaf clovers.