Any question answered...

Does the other frame have its own forks? Theres a significant chance of the old fork's steerer threads being 26tpi, in which case you would need new forks to go with the new headset.
If you are re-using the old forks, I would try again to get the old cups out.
The old steel is quite forgiving, use a long screwdriver and a hammer and dont be scared to really twat it.

Yeah, the two frames are identical (apart from condition) and I really bought this other frame just for the forks, which are the same colour.

I'll give it another go and then buy another headset if that doesn't work out.