As indicated above, casette would not work, as your hub is for screw-on freewheels. On freewheels, see
the article by late Sheldon Brown. 7sp freewheel should fit your bike, it would surely fit if the rear wheel spacing is 126mm, but I have squeezed 7sp freewheel on a 120mm Raleigh frame too.
Your gears are not indexed anyway, so no problems there.
8sp chain would work OK, Sachs/SRAM PC68 is the best (and looks nice, too).
I would start with getting the new inner tubes and tires (if they have disintegrated, but try first!) and a quick clean of the chain. It might work OK with old gears and all. After all, you have nothing to lose if you use the existing stuff. If it is all rusted solid, go ahead, buy a new chain and a freewheel.
Usually there is no reason to change the chainrings. If they are steel, they will last forever. If they are alloy, they will last well in excess of 30k miles (I just cleaned and checked mine). Getting the chainrings, apart from not being necessary, might me more difficult then finding a suitable freewheel. Chainring bolt hole spacing standards vary, sometimes they are not replacable at all, and then you will need to find the right set of (expensive) cranks to match your BB spindle. You will end up changing everything (trust me I've been through that path before, many times), better ride what you have!
Mateusz (ex compulsive carboot 15 quid-a-piece bike buyer)