You likely have a tank like one of theses below
formatting link
One of those tips at the top is where you pump air into the tank like you would on a car tire. The tank has a pressure rating on it someplace. Inside is a bladder that is filled with air. This bladder pushes against the stored water and this gives you water pressure when the pump is off.
Anyway, what may be happening is 2 things. First the rise and drop of the water pressure is when your tank empties and then the pump is sending the water directly to the house plumbing. If need, you can put air into the tank with tire pump. Just don't exceed the rating on the tank. Look for a number with PSI after it.
As for the hot and cold combining. Shut of the water flow into the hot water heater. Then turn on only the hot water in the lowest point in the house. This is usually the bathroom. Allow the water to drain. If you turn the hot water on in the bathroom with the water intake on the water heater off and you still get water, you need to check your shut off valves.
You may have sediment in the lines, hard water, or just age has warn out the rubber seals and washers inside the water valve(s).