There is no reason to run the 'heater' all winter long. Just winterize it and it'll be fine.
To cover or not is up to you.. I cover my trailer. But, it sits under trees while it's parked over the winter, so it's a no brainier for me..
I've owned this trailer since new and have parked it in the same place since new and never a leak or issue with it.
I'm only on my second ADCO cover too... Finally had to dump the 14 year old cover when the top had too much moss on it..
Anyway, I cover mine.. Others have indoor storage and others swear a cover will kill your RV.. So, just depends on what's gonna work best for you.
Cover has worked for me over the years. I will continue to use one as long as I park an RV on the side of my house, under trees.
This is the old cover about 6 years ago.. I replaced it 2 years ago with a new ADCO cover..

Good luck!
Mitch