Kind of Blued

I couldn't figure this out via searching.

I have a '79 Raleigh Super Course with the original wheels, and presumably, the original rear hub/freewheel. Until I can afford a modern bike, I'm looking to make this one more usable on the cheap. I'm wondering if I can use a 7/8 speed cassette in combination with the friction shifters and stock wheels without any problems.

If so, what kind of cassettes? This bike was made in the UK, so I'm concerned about compatibility. Are they Shimano compatible? Some of the 7/8-speed cassettes on the linked website are exactly what I'd like in regard to gearing (13-34). https://sheldonbrown.com/harris/k7.html#7cassettes

Also, will I need any other new parts, like a thinner chain?

Thanks!

Chombi

All depends on frame rear stay spacing.
126mm = 7 speeds max. you will need at least 130mm to accomodate 8 speeds. Of course you will also need the correct width hub to match up to the stay spacing.
You can cold set a steel 126mm frame to 130 to go to more than 7 speeds, as it's already been done succesfuly by many, but do not try it with a CF or aluminum frame.

Chombi

Kind of Blued

Also, I'd probably need a mid-long cage derailleur to get to the 34T on the back, right? I have a 14-30 on there now, so I may just go with a 13-30 to decrease the gaps between the gears.

Kind of Blued

Should I be measuring from the inside of one dropout to the inside edge of the other dropout? The best measurement I could get there (with a tape measure and conversion) is 123mm, but surely that's off by a bit. Is it safe to assume that it's actually 126mm, which would allow me a 7-speed cassette, or is it possible that Raleigh made chainstays that were closer than 126mm apart? (again, measured at the closest point between the dropouts)