Should I be nervous about my bathtub?

It would take YEARS for your bathtub to cave in, if its build and install right. i say this because it would take years for the wood to rot and become weak. if you dont use siliconized white caulking, or any caulking for that matter, the water damage below the bathtub WILL GET WORSE. Its common sense, water is getting past the tub and dropping down the walls and behind the paint...... all you need is a tube of siliconized white caulk, and apply a thick line around the edge of your tub and you should be good. be generous with the caulking, and dont use the tub for 24 hours after you apply the caulking, if possible. at the very least let the caulking dry for one hour.

Use a silicone caulk around the edge of the tub where the grout is wearing away. This will seal the gap and the leaking will stop.

Regarding the tub caving the roof in, I seriously doubt it. The discoloration is in the drywall, which is probably pretty soft if it's been soaking in for a while. The wood frame is what's supporting the tub. And, that would have to be really soaked for many years to rot out to the point where it won't support the tub any more.

Ideally, you should fix the leak, remove the damaged drywall, replace any damaged studs, put in new drywall, mud up the gaps, re-texture, lightly sand, wipe the dust off, prime, and paint.

The problem has progressed to the point the walls need to be taken down due to two Issues. First MOLD,Inside a wall is the perfect environment for mold to grow. 2nd is to see the extent of water damage to the floor and ultimately the support for the tub. CAULK cannot solve this problem.

The biggest part of your problem sounds like you have a major leak/s

to deal with first.When you can fix the leaks in the plumbing,then you have to take out all that previously wet walls,tiles,etc.anything that got wet(otherwise,it will rot)and then things really will start falling in.

Once you fix the leaks and make sure that you get them all,it'll be much easier to fix the other stuff,and use a waterproof sealant around your tub.

You're joking right ? if you ain't then buy a tube of silicon and seal up the gap or the stains will get worse till they're minging.

Yes you should call the maintenance guy.