Caramba Jeff :shock: :shock: A GA40 Les Paul!! According to Wally Marx' book (page 86), it's looks like a 1956:
In the '50s, a number of Gibson designs used pentodes ... built like a power tube ... as preamp tubes. Sometimes the 6SJ7 that had an octal base and sometimes the 5879 that's found in both preamp channels of your amp. Both Snake River Amplifiers and Divided by 13 Amplifiers make amps based on the 5879 as did Victoria Amps in their 're-issue' of the GA40 a couple of years ago.
That's a Jensen Alnico 5 field coil speaker .... that chunky thing hanging on the side is more than an output transformer. At some point, it became cheaper for speaker manufacturers to produce speakers with permanent magnets; the magnet in your amp probably has very little magnetic properties when it's turned off. When turned on, the power supply sends power out to that coil and, like a jHS science fair project, magnetizes the U-shaped chunk of iron under the bell. Unlike permanent magnet speakers that can be changed out like underwear, you either replace with another field coil speaker or re-engineer the power supply but the cone and voice coil are no different than any other speaker.
If you have trouble with the trem, chances are it will be that one or more of the caps in the trem circuit are shot. The trem function is 'on' whenever the amp is turned on but it's just grounded out when switched off. Those (probably) original caps in there have been capping away since the amp was built and, annoyingly, they are under that circuit board. Even at its slowest settings, you may find you'd like it to be slower. It can be slowed by changing the value of the caps however if you do it, it will never reach its original fastest speed; the speed would be cut at all pot settings.
Our member mad dog has a GA40 that's a couple of years newer than yours. The Tonequest article and Victoria re-issue pushed interest and prices for these up although the bloom may be off the GA40 rose ... with prices falling noticeably. Anyway, there are several back-in-the-day amps that stand out with their very distinct, signature tones; Fender 5E3 Deluxe, Vox AC15, Marshall 100W Super lead ... and I think the GA40 goes in that group too.
Congratulations!