Electrical – How to run chlorine feeder at the same time well pump is running

At 2.5 GPM, your 2-3 minutes of shower time between cycles is using between 5 and 7.5 gallons. If your tank starts at 60psi and 7.5 gallons is enough to bring it down to 40psi, that means you're starting out with 15 gallons of air in your 40 gallon tank, and ending the 3-minute cycle with 22.5 gallons of air. I calculated this using Boyle's law, or P1 * V1 = P2 * V2. And of course these numbers would be different if your shower head is higher-flow and your 3 minutes of showering used more than 7.5 gallons.

I believe to get the most output from your tank, you want it to be nearly empty of water when the pump is set to come on. That means draining and charging it to just below the pump setting, as you did. If you wanted to get a little more cycle time you could probably even go as high as 38 psi. Note that with a bladderless tank, you're going to need to recharge periodically as the air in your tank is gradually absorbed by the water.

For what it's worth I think 2-3 minutes is well within your pump's design cycle time. 15-20 seconds was way too short and by noticing it and fixing it you've probably saved yourself an expensive pump replacement.

All sort of issues here - while there's a short list of things that one normally looks for with extreme short cycling (such as a blown bladder in the pressure tank) that doesn't touch the 90 PSI issue...

If, in fact, water is going back down the well, the check valve in the pump (there is pretty much always one built into deep well pumps) is bad. Depending how your system is set up, there may or may not be another one inside the house, which would also have to be bad. My own system, after considerable reading and opinion-gathering, has only the check valve on the pump 300 feet down my well. When it goes, it's pump-pulling time. That would be the case even if there were others, and others might actually cause that to happen sooner according to a number of experienced well folks, which is why the other one I bought is still sitting in a box, not installed. In another few years I'll put a new pump on the shelf for when this one dies, so it's not a panic buy when (not if) I need it.

80-90 PSI is VERY high and almost certainly indicates a problem with the pressure switch. Certainly on my well the emergency relief valve lets go at 100PSI. On 40 off 60 is usually the upper range of domestic well pump "common settings" (30/50 and 20/40 being the other two commonly seen.) Ideally you'd know what your setting usually is or it may be recorded on a sticker somewhere on the system. You mention it going to the very high pressure when first switched on after being off for a while - what was it cycling between when you found it?

You could TRY hooking up a garden hose and opening that valve, then turning the pump on to flush out the crud that's been stirred up. It's possible-but-not-likely that this MIGHT clear a bit of crud that is jamming the check valve open. If nothing else it will get the stirred-up crud out of the well before the pump is repaired or replaced.

If your well is prone to throwing dirt, you may want to add a "spin-down filter" (I put mine after the relief, but before the rest of the system - this was contrary to the manufacturers instructions - they suggested after the pressure tank - but in line with the whole point of having it.)