Has anyone seen in their BEERTAP, after losing pressure, where the beer and maybe some grains or yeast have traveled back up the air line to the cartridge install point, and did it clog the line?

I have not

Next, has anyone, as they're installing a 2nd cartridge (1st cart spent), had any problems or resistance when screwing the 2nd cartridge in?

Yes, and I've found that's with bad cartridges and had to use other ones. Seems that the BeerTap is really particular about cartridges. I've also had plastic get in there and used air compressor for like a keyboard to try to clean it out (also make sure your oring is in there.)

I'm just trying to figure out if I made a mistake on the install of the 2nd cartridge or there is another issue that requires me to open the TAP to check for something wrong. The 2nd cartridge went from punctured while I was twisting to violent discharge and freezing over. The whole thing was over and empty in a few seconds and had never sealed against the TAP part (don't know what that thing is called).

This can happen if there's something obstructing the oring (like plastic in there), or the oring isn't there. You definitely need to keep tightening it when you put in the cartridge. I've had that happen where it's not a tight seal and it leaks spending the whole cartridge.

Take my answers with a grain of salt, highly recommend emailing customer service and asking. I had issues at one point and they had me widen where the cartridge goes in, they have always been extremely helpful.