As you found, a tub that is not installed level can create problems even if the walls around it are constructed properly. If the ceiling is soggy, it needs to come out and be replaced. While you have that out, you might be able to view things above (probably not, though). It should give you an indication if the subflooring is soggy and decaying.
You probably need to start tearing out at least a little of the wall around the tub. Depending on what is on the walls now, you may be able to reset the tub then repair the walls, or it may end up being a ripout and rebuild. You could have decaying walls and subflooring. You probably need to tear things out enough so it can dry out, then see what is still solid and salvagable. If it is solid, cleanup with some bleach and water should suppress any mildew if it can stay dry afterwards.
Are the walls around the tub tiled or paneled? Do you know what is behind them? If tile, are they soft or well anchored to solid substance?