The formation of ice requires at least 3 things.
- (The normal, linear passage of) Time
The time factor is clearly seen in both Einstein's short trip and Marty's trip from 1955 back to 1985 at the end of the movie.
Here is the return of the DeLorean with Einstein just as the car is spinning to a halt.
But here it is a few moments later.
Here is the car just moments after it has crashed on return from 1955 to 1985. Some mist, but not much ice, if any.
And by the time the terrorists are coming to get revenge on the Doc., ice is apparent.
So the Doc. mentioned cold, we've seen the effects of time, the other factor is moisture, or in this case, humidity.
The scenes where Marty goes from 1985 to 1955 are slightly different, in that the car is surrounded by mist, but there is little evidence of ice on it while in the barn. By the time Marty drives it back out of the barn, the effect has vanished.
Here is the first part of the time in the barn, the mist is obvious but there is not much ice, if any.
A few moments later we see that the mist has cleared, and though there is no ice on the car, there is a sheen of moisture on the bonnet and windscreen.
My conclusion is that in this trip, it was a lot less humid (to gain less ice), & significantly warmer, to melt it faster.
I was going to review the other movies, but I think this explanation covers why the ice might vary according to the trip.
DVK covered the other question, the flying circuits were destroyed by the bolt of lightning that hit the car at the end of Back to the Future 2. Here is the relevant part of the letter read near the start of BttF 3:
Doc. Brown: The lightning bolt that hit the DeLorean caused a gigawatt overload which scrambled the time circuits and activated the flux capacitor and sent me back to 1885.
The overload shorted out the time circuits and destroyed the flying circuits.
Unfortunately the car will never fly again.