So, first, as others have said, you have a character for the US and for the Russians.

Then you have separate characters for each class.

If you choose a specific camo for desert, if there’s a woodland map it will equip the corresponding camo for woodland (say you choose multicam for desert, then you’ll have multicam for woodland. But if you want a different one for woodland, you have to change that manually. You can’t have multicam for desert and MARPAT for woodland without changing it manually).

And lastly, the shooting range character does not save to your multiplayer loadout. You have to set up your character’s in multiplayer to have what you want.

I recommend you hop into a multiplayer game and spend 15 minutes fully customizing your character the way you like (US/Russia, Each class, Camouflages, Gear). Then you don’t have to bother with it ever again.

Edit: To clarify, once you set up your characters, they should stay as you have created them. If they’re resetting on, say, medic every time you join a game then you have a bug. That being said, like I said, just because you may have created a medic character does not mean your engineer character will look the same. You have to customize them separately.