The Elder Scrolls V Skyrim – How to Skip the New Game Introduction Cinematic

Yes, it is possible. The main quest is not completely independent of the war, but it does not require your character to take any action to support or oppose either side in it. You won't need any advice or hints as to how to avoid taking a side; the game will tell you what you need to know when you get there. I hadn't realized it before, because I wasn't in this situation myself, but some thought seems to have gone into making this work out.

Frost only becomes your horse if you persuade Louis Letrush to let you keep him or kill after you talk to him after you have stolen Frost from the Black-Briar Manor. If you just ride off without completing the quest Frost will just return to where you first stole him (Black-Briar Manor).

If you own him and you were to mount another horse (at least another horse you own) he will remain at the location you left him so far as I can tell. I use Shadowmere (who seems to do the same thing) and I just parked Frost at a stable so at least he would be safe. He has stayed there since.

Also any horse you bought from a stable will return to that stable when you begin using another horse. When I first got Shadowmere the horse I purchased from Whiterun began to walk away and when I next went back to the Whiterun stables he was there safe and sound.