Mine has been plugged in and stored outside beside my house since October and will remain plugged in until my first road test (May) before a camping trip in the spring. Temps here were unusually cold in December (-30C/-22F) for a long time. Then had a thaw near end December to 12C/53F and I checked the batteries. Everything was fine. All appliances inside are turned of/winterized. Another thaw coming this weekend and I'll check everything again. Have done this will all my MHs with no issues.
As for power surges (we get a lot in our rural area) my surge protector takes care of them. As for cost, I'm guessing a very few dollars a month.
My previous MH did havev an issue and would not charge the chassis batteries when plugged in and after freezing them (and replacing) I put in a Amp -l-Charge. Solved the problem.
Chris