Thanks for all your help! Much appreciated.

The wobbling rear end looks to be pretty bad - bad enough that when I put the bike on a repair stand and just spin the rear wheen freely, the wobble is enough to start the cranks rocking gently up and down - the chain transmits the wobble to the chainrings which start rocking along.

I'm pretty sure the rear sprocket set is a freewheel type - the bike originally had six rear cogs, and the rear hub itself looks to be original.
What is all that I have to change to fix this issue? Furthermore, as it may take a few days to get all the necessary parts, is it OK to continue riding the bike normally even
with the very noticeable wobble?

Thinking about my list of upgrades, originally I thought I'd like a full modern groupset, but seeing as there are many other options as well, I might just reconsider.

How about fitting a Tiagra or new 105 front derailleur? Shouldn't that make averything work smoothly with my new brifters and no need for excessive fiddling?
And leave the rear end as it is, just fix the freewheel/hub.

I feel the brakes are enough for my use as they are, but like mentioned before, new ones would be cheap as well.

My main new question is how would I go about getting wider choice of chainring ratios? I feel the small front ring (a 42) is a bit too big for me in relation to the
28 rear large sprocket. I have some extra body weight that I need to trim off (by riding this bike, no less) and steeper hills are hard work for me as it is.
I think I like the Biopace design on my chainrings, but I feel I need smaller ratios available to me.

Would it be possible to fit more widely-spaced rear ratios (7 speed, current cogset is a 28-14), or would my only option be to change the chainrings?