You likely have a tank like one of theses below
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.
bladder pushes against the stored water and this gives you water
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
sending the water directly to the house plumbing. If need, you can
into the tank with tire pump. Just don't exceed the rating
on the tank.
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
point in the house. This is usually the bathroom. Allow the water
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
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).