You most likely have a problem either with the tank's gasket system (water keeps leaking back to bowl) or else the shutoff valves are faulty (poor water transport). You could also have trash either in your shut off valve or in your fill valve in the tank.
The toilet flapper stays up to allow water to flow from the tank into the bowl when the toilet is flushed.
The toilet did not add water yet. By the way, If you filled a bucket with water, The water level would go up.
Gravity
The toilet flapper may not stay up due to a worn-out or misaligned flapper, a problem with the chain or handle, or issues with the water pressure or tank level.
If you are referring to the water closet fill valve, then you probably have a bad seal. most Ball Cocks fill from the bottom and have a feed line to the bowl. If the seal in the top of the valve near the ball arm is bad you will get water out of the top of that valve. If the line (usually a small hose that goes in to the top of the flush valve) is off it, will also spray water. sounds like the fill valve is bad and should be replaced. A fluid master 400 will fit most toilets and is easy to install. The amount of water in the bowl after flushing is very important to a good flushing water closet. if this hose does not stay in place the toilet will flush slow and clog up more.
no, provided there is not a high water table. It can stay empty for 2-3 months
because its tanks fill up with water.
in the toilet
No, but it can help. Just remember to stay near a toilet!
Yes, a water bottle can float in water if it is empty or partially filled with air. The air inside the water bottle creates buoyancy, allowing it to stay afloat on the surface of the water.
Because there's never any change in it.
There is no weight in an empty balloon. Empty balloons will not stay on the ground long enough to have any real weight.