If the tub has a shower as well, the pipe from the faucet to the shower head could be leaking. You might also check the tube that comes OUT and fills the tub. It could be leaking back at the valve when you turn on the water.
Yes, most definitely!
What is above your ceiling. Chances are it is a leaking pipe (probably a waste pipe-ick) or condensation from around your air ducts. Is this a house? Leaking on 1st level or a basement? Is it a "drop" ceiling?
lots
It means it is time to call a licensed master plumber to make repairs and possibly get ac mold removal /toxic clean up crew of to your abode
For the ceiling to be damaged there is a leak! As to who's fault it is that is the question. It is determined by who is responsible in your apartment for the upkeep of what is leaking. Some of the causes could be a leaking tile wall or tub surround, water leaking where the valve or spout protrudes through the wall, a drain leaking, the shower valve or piping leaking, water spilling over the edge of the tub on the floor or even a roof or window leaking. I would recommend hiring a plumber to investigate and repair any leak that is occurring as most of the time it would be your responsibility for repair. A leak is not going to get any better on its own and water damage costs can be significant.
Ceilings do not leak as there is no water The piping arrangement above the celinging or grouting may leak and come out through the ceiling
Yes, kitchen faucets do waste water, but in very small quantities. It won't cause catastrophic damage to the environment if you don't get it fixed, but, it'll probably cost a good deal in repairs sooner or later.
It will cover the building but not the contents.
By finding and repairing the leak.
A drain leak, a leaking shower pan, a leaking shower valve or piping, a leaking bezel or escutchions on the shower valve, bad or missing grout, bad caulking in the corners of the wall, a leaking shower door, water spilled on the floor, or, a drain pipe leak. any one or combination of the listed problems can do it. Look things over take care of what you can see wrong before making a hole in the ceiling. If you think you have fixed the leak repaint the stained area but prime 1st with Kilz or Binz to cover the stain or it will reappear through the paint even if the leak has stopped. If the leak hasn't stopped from taking care of the things you could see then open up a hole to see where it is coming from.
Mould is mostly caused by humidity or leakage. For example, if you have mould on your ceiling, you should check if your roof isn't leaking.
NO -- you will have to open up wall behind tub to see where it is leaking from.