Jason,
Most of your system description I understand, except for the '32kb expansion module' and the 'PC Enterprises 1.2MB parallel port (I believe) expansion via sidecar'. Is the 32kb expansion module the memory card in the machine? If so, that's actually a 64K card. Is the PC Enterprises option an external floppy drive?
Barebones on a Jr is actually 64K, which is soldered on the motherboard.
Floppy drives are always a source of problems on an old machine. I recommend cleaning the drive first. In another thread I posted a link to a PDF file of the procedure that the Eugene Oregon PCjr club used to distribute. Basically, you clean the drive rails (what the drives head carriage slides on) with isopropyl alcohol and then you lubricate those rails with a light machine oil. I use a tube of oil with Teflon particles in it that I got from Radio Shack - it's expensive, but a little works very well. (On anything you use apply it with a Q-tip and never spray anything .. you don't want to contaminate the heads.)
The next step is to clean the drive heads. They are delicate, so this is best done with a dedicated 5.25" cleaning diskette. If you do not have one available it can be done with Q-Tips and isopropyl alcohol, but you need to be extremely careful to not knock the heads around and out of alignment. Also, you don't want stray pieces of Q-Tip cotton being left on the heads. If you do this be very gentle.
I find that normally fixes drive problems. The other thing that might be wrong is that the drive belt on the bottom of the drive might be stretched or worn and causing the drive to spin the media at the wrong speed. There is some tolerance built into the drives/media for drive speed RPM differences, but if it goes too far the diskette drive won't be able to read or write diskettes not created on that drive.
After you clean the drive get a blank diskette and run the built-in diagnostics. That should give you a good idea on if the drive is usable or something deeper is wrong.
Problems with the floppy controller are very rare - there are no moving parts to fail. Don't bother with the 1.2MB drives - the distance between tracks is 1/2 what it is on a 360K drive because 360K drives have 40 tracks and 1.2MB drives have 80 tracks. The first track will be readable, but nothing else. You will actually have better luck with a 1.44MB floppy drive, as those are always 80 tracks whether they are 720K or 1.44MB drives.
On the keyboard problems ...
You do have batteries in the keyboard, right?

Nice fresh alkaline batteries installed correctly, or nothing works. If that is not the problem it is possible that the lighting in the room is too bright, or that the IR sensor has been shaken loose on the motherboard. It is a separate module - just press down slightly to ensure that it is seated and that the receiver is aligned with the hole in the case.
In general be careful when handling the cards and pieces. The pins are easy to bend, and that is a pain to fix. Proper static protection should be used too. Dig out the 'Guide To Operations' and run the diagnostic procedures .. test every possible option to isolate what is wrong.
These things are hard to 'wake up' after an extended slumber. Don't let it sleep that wrong again. ;-0