I have an old No. 5 Remington Rolling Block. It's a 7x57 military rifle, with a barrel that squeezes a round ball down to about .287 at the breech and again at the muzzle, but for about a foot somewhere in the middle, the slug just drops.   

I'm not a hunter, but I'd like to make the old roller into a something with a shorter barrel.  There is a lot of .223, and especially 5.56 mm military stuff around.  Would that work?

Those are the "stronger" of the roller actions but you need to look at pressures and make your decision. I am sure someone here can help you with that. I rebarreled one with a Douglas barrel a few years back and he went with the original 7x57 round.



JMH

Yes, the extractor should work with only minor stoning with 30-30 head size cartridges.

Yes
on a .43 whatever action, rotary extractor
firing pin is a little large, not a problem if you keep the load sensible
I also have a No.5, with a sliding extractor, original 7mm, barreled to 38-55, had to make an extractor, wasn't comfortable welding the original
keep in mind both no.5 were set up for rimless
also mine is a 1897, with the large barrel threads

Ok, but how about with a 1902/1910 action with rotary extractor? That's what I have to work with.

welded the extractor and fitted to the 30-40K rim
easy enough if you know a good welder
fitting to the barrel requires a lot of patience

I've tightened up #2s by making new block and hammer pins from drill rod, hardened and tempered after fitting.  The same thing could be done to the larger editions of the Rolling Block.

A "real" fix would doubtless involve reaming the block and lever holes round and then making new pins to fit the larger holes, but just a couple thousandths more diameter in the pins helps a lot.  Before the change, I could pull the blocks back against the hammers and see daylight between the block faces and the barrel breech, but afterwards this wasn't possible.