Retro Bike Restoration

Be very careful with which way around you put the front wheel in. You can see when you take it apart to clean and regrease the hubs, the RIGHT hand side has a stop which the cone is screwed up tight against. The left doesn't. If you put it round the wrong way, the bearing cups can tighten up and crush the hub. I speak from (juvenile) experience!

Plenty of good advice here. For the spoke length I would recommend removing a spoke and measuring it. The length is determined by measuring from the inside of the bend. Be aware that there will be two different lengths, the gear side and the non gear side, although the broken spokes are most likely on the gear side. If you're buying the spokes from a local bike shop you can just take along the one you've removed from your wheel.

One thing I found when I restored my '84 Claud Butler - the chains are cheap as chips. I've still got a spare, boxed one in my spares box - if you're in the UK I'll chuck it in the post for you, I'll never use it now.

The 28 will allow you to have easier gears going uphill, a higher cadence is more efficient, the 52/14 combo is your high gear, that'll be fine.

The RX100 is a good solid mech, you might need to screw in the B screw so that the upper pulley will clear the teeth on the 28 sprocket but otherwise nothing else really aside from normal alignment adjustments.

Yes, where the spoke comes out the hub hole, it'll probably be about 295-300mm as a guess.

Any 5/6/7 speed chain will work fine, you do not need (nor want) a hyperglide type. A bog standard KMC Z51 chain would be fine and reasonable quality.

I had a Raleigh Equipe back when I was at school before it was pinched, did me for getting to school and out and about. Moved up to a Falcon with my paper round money

Couple of things.

Is the freewheel seized/not spinning up, what is wrong with it that you need to replace it? It doesn't look like it's had much use and a squirt of liquid grease into the pawls might be enough to make it usable/spin more freely.

Is your replacement 14-24 going to give you a low enough gear ratio for the cycling you will do? If you are sticking with the 52/42 chainrings you might consider a  14-28 which will give you a lower gear range.

As above, there's nothing wrong with the chainset, the inner ring isn't actually worn at all, that's how those older teeth look, if the bottom bracket is okay and not loose then you don't need to do anything though bearings may well need regreasing in any case which would mean taking the chainset off the axle.

Spokes, measure from the knuckle of the spoke to the inner of the wheel rim then add around 4-5mm.

HTH 

Agreed, those rings look absolutely fine.

The freewheel and removal tool should both be fine. As for the chainset the large 52t is integral with the chainset so cannot be replaced, the only option would be a replacement chainset. However the teeth don't look particularly worn so I see no reason to replace it. Good luck with your project.