When I turn the water on, full force, at the faucet, it runs fine for 20 seconds or so, then the well pump kicks on, and the flow drops to a near trickle. Upon shutting the water off at the faucet (with the pump running), it takes about 5 minutes for the pump to pressurize the tank. Bad foot valve? Clog in pump somewhere? Clog before or after pump? I'm sure the pump runnig is costing me money, but more important, It shouldn't take 15-20 minutes to fill the washing machine. I have scoured the internet for this exact set of symptoms and, as usual, I have some kind of "unique" problem, which no one has ever asked about before. Somebody help me, PLEASE?

No mystery here. You open a faucet and use the water stored in the tank in 20 seconds. After that, the water you get is being delivered by the pump at the rate the pump which it can pump it. After you turn the faucet off, it takes 5 minutes for the pump to refill the tank.
Why would the amount of water delivered by the pump be so low? Because the well is going dry or the water level in the well has dropped. If the pump is not at the bottom of the well you may be able to lower the pump.

You need a well man to assess the well.

I think HK is right. If you want to be sure, run an outside faucet until the pump cuts on, then run water into a 5 gallon bucket for one minute. See how much water you get in that one minute. It should be a minimum of 4 to 5 gallons. I'm going to guess your's is nowhere near that.

What size pressure tank do you have?