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Just so I understand, the pump only operates when you manually trigger the pressure switch?
Using a 5/16 (might be 3/8) nut driver, can you adjust the spring tension on the pressure switch to determine if it will trigger at all?
Also, do you have any way to shut off the line so that you can isolate just the pump equipment from the rest of the house? (I guessing there is a leak of some kind, but where is a question)
Do you have a model number for the switch?
The only time I've seen a situation like yours, is when the pressure switch gets debris in it. I strongly suggest the low pressure and bleed down are a seperate issue, probably a leak in your line. Most common spots for that are on the verticle line running down to the pump, and any couplings/fittings in the line.
As far as debris goes, if the pump sits close to the bottom of the well, it can suck up sand/etc... or if you have galvanised piping, rust could be an issue.
The pressure switch not asking calling for water is where I'm thinking your debris issue it. If it was in the switch itself, it could keep the switch from actuating. ie calling for water. If you can trigger the pump on and listen down the well head, you can many times hear water spraying in the well shaft (inless the leak is under water) the buried section will need to be dug up. Simplest way to determine these issues is to pull the pump and air test the lines.
very possible, I've seen the rubber on those go bad.