Your system sounds like it is is fine. The inverter is supplying power in the absence of power from the pedistal or the generator. The transfer switch controls the flow of A/C power (either pedistal or generator) to the coach. You should turn off the generator before you connect to shore power and also disconnect from shore power before you turn on the generator. If you don't, you can easily damage the transfer swtich.
The problem you had was caused by your transfer switch not transferring power back to the coach electrical system. Lots of potential reasons for this but if you want to try a few things yourself you can check the wiring inside the transfer switch for loose wires (do disconnect all sources of power before you try this). Many times there are loose power leads inside the transfer switch that just need to be tightened down. Loose connections mean heat and intermittent connections - just like the problem you were seeing. Also, if it happens again, give thhe transfer switch a good thump with your hand. That can free a stuck relay and get it working.
I don't know about your frig, however. Since the freezer is at -4, it is obviously cooling. The refrigerator section being at 48 doesn't sound right. I'd give Samsung a call and ask them what is wrong.
As far as needing anything, I suggest you have a Progressive Industries HW50C EMS system installed before the transfer switch. This will protect the coach from miswired pedistals, too low or too high voltage, open grounds, etc. Think of it as good protection for all of your electrical goodies.
Roger & Mary
2014 Tiffin Phaeton 36GH
2012 Honda Fit toad