Be careful disconnecting bypass capacitors. The mfg. put them in there for a reason. Many times it's to prevent the tube from going into self-oscillation or to bypass any RF that could be picked up by the wiring or other components. AM broadcasts don't go beyond around 8-10 KHz in frequency response anyway. The amp was probably designed to make the broadcasts of the day sound good. There was no "hi-fi" back then. As far as the trimmer cap is concerned, if you know which band it's associated with, tune in a station near the high end of the band and adjust it for the loudest signal. Don't adjust anything associated with L1, L2, or L3. They are oscillator tuning coils and caps and determine the frequency range that band receives. As far as the speaker is concerned, you can try glueing the cone back down to the frame. Try to keep it centered as best as possible. If you have (or want to) replace it with something more modern, be aware that the schemetic shows it has a "voice coil" arrangement. They used a coil in the speaker housing as a filter choke in the power supply, which also doubles a magnet for the speaker itself. If you replace the speaker with a standard one, you'll have to replace the filter choke winding with a fixed choke or a resistor, maybe a 470 ohm 5 watt type. Sounds like a fun project.
Rich