I just bought my 2005 accord a few weeks ago, 2.4 liter manual coupe, used, 99,999 miles

and I'm finding out it's funky problems...

when it's cold out, the radio comes on by itself when I'm getting gas, or the radio won't play any sound for awhile or until I turn the car off and back on again sometimes...

Well today, Got in my car and the time clock was already lit when I got in ... I had to get in with the key since the fob wasn't working all of a sudden after work & it is 18 degrees out today...After shutting it off while stopping at a store, my radio came back on this time when my car started, but my fob is still dead, still had to use the key to enter. The fob was working halfway across the parking lot when I originally bought it....

Is this the cold that killed something? Is is just a dead fob battery on coincidence? My bf checked....the power door locks work internally and with the key...Door switches are okay...trunk won't open by remote but opens by inside latch...wtf is going on with my NEW used car already?
Please help.