Usually cases of a slowly decreasing amount of water are that a filter has become clogged. Does your water supply have sand particles coming through? Check the diverter heads for other faucets in the house. Are they getting many small particles trapped in them? Sand in one faucet tells you that all fixtures are getting clogged, some more slowly than others. This sandy junk can come through municipal water systems, or it can come from your own well if you are on one.
If you are getting much of this coming through, you may choose to install a whole house water filter to prevent that.
Whatever you find in those other faucets, if the problem is both the shower head and the tub spout, then the problem is in the fixture itself. Is this one of the new style fixtures that regulates the temperature of the water? I have read that those temperature regulators can sometimes be a problem themselves, improperly shutting down water flow.
If not that, assuming this is a cartridge based fixture, then you will need to pull the cartridge from the faucet. This entails removing the handle, then removing a locking nut that holds it together. Please shut off the water BEFORE you do any of this! The cartridge will come out, although this sometimes takes some amount of persuasion.
You can buy either a complete cartridge replacement from most manufacturers, or just a set of new o-rings etc., to rebuild the cartridge. Large home centers will have an entire wall filled with various such replacements. Bring in the old cartridge if possible to get a match. Knowing what company made your fixture will help.
A possibility is that a piece of rubber from an o-ring or other seal has cracked and become lodged in an orifice inside the cartridge. Look for this, or for other problems when you open up the cartridge.
Hmm. I'm going to guess this is possibly water-hammer related, where a high-pressure spike caused by the toilet fill valve shutting is forcing water past the mixing valve. If so, the water would be released as the toilet stops filling, and a water hammer arrestor should settle it down, be it new and high-tech with a piston or old (merely an air-filled pipe stub that needs to be drained of accumulated water from time to time.)
If it releases water at the start of the flush, I'd have to suspect some sort of vacuum breaker being involved, and getting activated by the sudden drop in pressure when the toilet fill valve opens.
Edit: ahh, your new information is possibly diagnostic - turning the hot water supply off stopping it suggests that the casue is probably related to the "anti-scald" function that's pretty much universal in tub/shower valves at this point, which is responding poorly to the pressure drop on the cold line as the toilet fills (and for some reason, doing so even if the mixer is shut off.) You might call the valve manufacturer and inquire if this is expected behavior or a defect of some kind.
If it is expected behavior, and you cannot run a new line that's either larger (for less pressure drop_ or dedicated (so the toilet and shower are not sharing the line) a small pressure tank/accumulator in the bathroom area should sort the issue by holding pressure up while the toilet fills.
On third thought, first go though and make sure that all the valves in this line are FULLY open. If the problem persists, see if your toilet fill can be adjusted to be slower. Then consider more involved solutions.
Best Answer
The circumstances that you have related seem to point to the shower head being placed on the floor as the cause of your damage. To rule out other causes it would be best to duplicate all possible scenarios (e.g. fill the tub and check for leaks; use the shower normally and check for leaks; lay the shower head on the floor and check for leaks, etc.)
NO, unless purposely built as a "wet room" with a floor that slopes to a drain, bathroom floors are not waterproof and significant flooding will damage underlying structures.
A single event might cause such damage but only if it was a significant event and there were conditions that prevented drying, and it would take a long time. That type of damage seems to have been caused by repeated events over time that kept the area moist.