How to calculate water pressure loss due to elevation change?

If there were no frictional losses then the pressure as the water returns to the tank will be the same as the pressure where it left the pump. The only energy needed is the energy to accelerate the water to the speed necessary to move along the pipe. You cannot look at the pressure at the top of the pipe unless you are interested in cavitation which will not occur until a vacuum is created at heads of greater than 32 feet. The pump will not be pumping 14 gpm unless the pressure AT THE PUMP is zero. And this is not the case. The flow MUST be the same at all points along the pipe ( series circuit rules) You may not calculate different flow rates at different points along the pipe unless you include a storage tank at the top of the pipe. As your units are so awkward ( gallons per minute, psi, inch pipe) the conversions are too awkward for me to do for you. Incidentally if you use an exponential horn at the end of the pipe in the tank then the kinetic energy is used to lower the pressure which in turn makes less pressure required at the pump.

In the situation you describe, the only pump pressure that develops would be to overcome friction. You would not need to even consider the pressure difference due to height, as the pressure will be equal on both sides of the pipe and offset each other (talking only of pressure due to height). Liken it to a siphon pipe if you like, where the only flow is due to DIFFERENCE in height due to difference in reservoir head and outlet height, which needs to be lower than reservoir height for siphoning to occur. Height has no bearing on the function (other than increased friction), therefore flow rate). The only requirement would be the downcoming column of water would need to be solid (no air). That COULD be difficult to maintain. Smaller pipe could be advantageous to purge any air out, if the sched. 40 you suggest does not have the velocity to do so. But, pressure would rise, and flow rate diminish. I'd suggest the only practical way to determine what is required would be by experimentation, unless you could find someone who has practical experience at similar systems, such as a swimming pool technician.

In the situation you describe, the only pump pressure that develops would be to overcome friction. You would not need to even consider the pressure difference due to height, as the pressure will be equal on both sides of the pipe and offset each other (talking only of pressure due to height). Liken it to a siphon pipe if you like, where the only flow is due to DIFFERENCE in height due to difference in reservoir head and outlet height, which needs to be lower than reservoir height for siphoning to occur. Height has no bearing on the function (other than increased friction), therefore flow rate). The only requirement would be the downcoming column of water would need to be solid (no air). That COULD be difficult to maintain. Smaller pipe could be advantageous to purge any air out, if the sched. 40 you suggest does not have the velocity to do so. But, pressure would rise, and flow rate diminish. I'd suggest the only practical way to determine what is required would be by experimentation, unless you could find someone who has practical experience at similar systems, such as a swimming pool technician.