I then wired the output of stage 1 to the input of stage 3. Playing the clean en overdrive section seems to be without any problem.
Obviously, the problem is somewhere in the part of the circuit that you jumped over.
I then wired the second stage back in. The amp works, but after a few seconds the volume drops. When I hit the guitar hard, some distorted sound is heard that dies out very quickly.
That suggests that the problem is
probably either a bad solder joint or a bad resistor, both of which can act like a bad/intermittent connection depending on how you push or pull on it. What's likely happening is that the bad connection (solder or resistor) is making good contact when the amp is cold, but as things heats up it pulls the connection apart.
I would reflow all the solder joints in that part of the amp, adding a little fresh solder to each joint as I go and making sure the new solder flows well. The new solder will help fill any dry spots, but most importantly it introduces a little fresh flux.
It that doesn't fix things, then start tapping on the resistors and caps with something insulating like a wooden chopstick while strumming on the guitar. It's helpful to have someone else do the strumming, but I've done both and operated a kick drum at the same time.