Is there a certain way to best handle modules?
I woiuldn't touch the board unless necessary... tube amps are robust, but there are some ICs and such that could be damaged by static electricity. Never hurts to be careful.
A way to insert them?
I hold mine by the front panel and start them on the track, then push until they seat, then use the thumbscrews to secure them. If you're having trouble making one line up/fit, there's a good chance that the screws on the 'cage' over the tubes are too tight. Loosen them a bit and it should work just fine.
A state the amp should be in when adding/removing a module?
At minimum put the amp on standby, but I turn my amp off. I wouldn't pull tubes with the amp on standby, so why would I pull TWO AT ONCE on a preamp module?
A way to store them when not in use?
The Randall box is great, but mine has been ordered for about 2 months now and I haven't seen it. Bought another one used on ebay. I'll be a happy camper when it finally gets here.
Best way to swap tubes?
I would brace the tube board (that is secured to the module by two screws) with one hand and work the tube loose with the other. Some of the modules I've worked on were CRAZY tight, but some 12ax7's seem to have slightly different pin sizes too. I would not recommend just grabbing a tube and yanking - support that PCB where the tube socket is located with one hand for sure.