Yes - as you have found you can save anytime with SH5
Just beware that if you save and your target is damaged and sinking, but not yet sunk - it will be 100% health when you load that save.
No damage to any ships are saved with the game.
Your base may have moved - look at wilhelmshaven for the sub bunker icon
No progress bar is odd - see if you get credit when you are back at your new base