In your pic is a modern hyperglide hub which is the current standard.
You may have bought a uniglide hub which is archaic and defunct.
https://www.sheldonbrown.com/k7.html#uniglide

Sheldon's suggestion was to transplant a hyperglide freehub on to there which is pretty straightforward.

I know there are cassette purists, but your 105 set is workmanlike and doesn't really need a fuss over period correct consumables. There are 2 off the self currently available decent silver 7 speed cassettes, the 11-28 shimano acera one, and the 12-32 sram one. Both about $20. I prefer the Sram.
Wider gears is always better. You would almost certainly shift the 11-28 and might do the 12-32 with a flipped b-screw, if not, a different derailer. The 11-28 can be improved by swapping a 12t top cog, so you get 12-13 (and a silver lockring).