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.
You're sorta kinda in a bit of trouble here. This system is close to being good, but because of a seemingly minor screw-up, you're going to have major problems here. The recirculating pump SHOULD NOT be connected to the drain.
First off, it's quite possible that the segment of pipe between the pump and the "tee" attached to the drain valve is completely or partially clogged with (corrosive) scale/sediment buildup. Judging by the scale buildup on the outside of the pipe just below the tee, it looks like the tee is leaking (why? corrosion?). I can only imagine that the inside is worse. Even the check valve and the ball valve upstream from the pump look like crap. In fact, I'm pretty sure that's calcium on the ground below the check valve (thing between the pump and the drain with a hex bolt -shaped top on it), indicating that it too is already leaking. The check valve almost certainly isn't functioning properly (check valves are intended to only allow water to flow one way,) which is possibly why you are having to constantly readjust the temperature in your shower. The cold water gets injected into the tank at the bottom (via a long "dip tube" that enters at the top and travels down to the bottom.) If this check valve is being held open by sediment, or if it has failed due to corrosion, then cold water could be coming out of the water heater from the drain at the bottom, making your "hot" water turn merely "warm" after about 10-15 gallons of use, which is one possible explanation for your "shower get[ting] colder during use" problem.
My advice would be to get this junk removed ASAP before something bursts and you have a disaster on your hands. Get rid of it completely, or if you want to get it fixed up, get a new pump (Grundfos makes great pumps, but I can't imagine that this pump is usable at this point) and tell the plumber to connect the recirculating pump to the T&P relief valve inlet, which is not currently being used(?), rather than the drain. Doing this would reduce the scale/sediment buildup greatly, which happens mostly near the bottom of the tank. Replace all of the pipe in the vicinity of the pump as well, as it is probably completely scaled up and corroded. Everything else should be ok.
I see a line (the CPVC that goes into the wall) for what I hope to be a T&P valve, but I can't see where the actual T&P valve is. If it's tee'd off of the hot water outlet, then that's fine. I'd hope it's not simply laying on top, not connected to anything, but I don't have enough faith in whoever installed this to take that for granted.
Best Answer
To set this up with a submersible pump, you need to have:
Optionally, you can install a second normally closed float switch (set to turn on when facing down) at a slightly higher level than the other one, and wired to an indicator light in the bathroom or wherever. This will alert you when the tank is almost empty.