It's impossible for a meaningful static charge to build up on a conductor of that sort. We are talking about a antenna cable here? The charge should be uniform as it is a conductor, and the shield is grounded so there's no free electrons to build up a huge charge. Of course antenna cable has some capacitance as it has two conductors and a insulator between them. But the amount is so small, that there's no way for signal generated static charge to have anything to do with throughput of a cable modem.
I don't think that cable companies and tv stations go around re-attaching cables constantly.
Sometimes, especially for humid enviroments, re-attaching cable ends will make a difference, as the wire connector will accumulate rust even when it is attached (it is not hermetically sealed). When you remove connector and re-attach it, this miniscule layer of rust will be wiped away -> everything works again.
But for the orginal poster - the AC adaptor makes noise? Might be normal, might not be. Try with a different AC adaptor (make sure it has enough amps!!).
The title says you drop your connection if used "for a long time"? How long, hours, days, weeks, months?
And some devices will get incredibly hot, and they can handle it. But usually hot is bad, and at least it will shorten the life span of your modem. If warranty is already gone, and you know what you are doing, check the capacitors. If not, do put the box somewhere more less hot... if floor is cold, put it there and as said - sideways.
Once I had a modem which did drop the connection especially when in lot of load. It had two bulged caps, changing them did some good but still I'd get drops. Things changed for the better when I ...upgraded the ventilation