1) I'd stick with the 105. It will work just fine with 7-speed.
2) 7-speed can be either cassette or freewheel.
3) With 126 mm O.L.D. you can just slip a modern 130mm hub into it. Or cold-set it, if you want. 120mm I'd definitely cold-set to get to 130 mm.
4) I've cold-set 4 bikes so far, and none of them have had any issue at all many thousands of miles later. Just go slowly and check alignment- one bend is OK, but you don't want to have to bend it back.
5) 7-speed cassettes work fine on a modern 8-9-10 speed hub. Just make sure to get a 4.5 mm spacer, since a 7-speed freehub is slightly shorter. 7-speed cassettes are widely available and will be for some time. 7-speed is all I ever use on my touring bikes- everything is just more durable and shifts more easily with friction shifters.
6) The bike might have compressible housing, which may be part of why the old derailleurs need constant setup. The new shifters will have new incompressible housing included- use it. Or the 105 stuff could be worn out, in which case you will need new derailleurs after all. But I'd try the 105 stuff first since it's way better than Tourney.