The thinner stuff you can get at any houseware store, or even the dollar store. It won't melt.
For turning the eggs, I mark one side then just turn them 1/3 of the way 3 times a day. I even use the tray and push it along since some of the eggs turn well (I'm lazy!).
Yes, I indeed have one and have the same issues as you. But, it can be rectified by placing some non-skid shelf liner underneath the tray. I didn't do that with this batch, just been hand turning them. The unit keeps the temperature well. I'm going into lockdown with my eggs tonight.
But...