I'll chime in for this one 😉
Definitetely do have an interest in unusual action types, one that I'm currently quite fond of is the rolling block.
Just a really vintage and cool action type(late 1800s) and with more of a safety-in-the-design approach than modern firearm safety designs(both safe by design along with a manually activated safety catch).

It isn't a particularly practical gun but it is a very enjoyable action to work.

And pics here:

And the nice thing is the rolling block action has a ton of chamberings, with varying commonality.
I've seen 357 magnum, 45 long colt, 44 magnum, 22, .17hmr, 16 gauge, 28 gauge, 20 gauge, 45-70, even one in 380 ACP which is an odd cartridge to chamber in.
I have one in 22lr and seriously contemplating one in 357 magnum 😁

I do know a guy who hunts pretty regularly with a combination gun, one barrel is a 20 gauge and the other is a 22 hornet, and it does pretty well for almost any pest he encounters around the farm when he's on his tractor.
No pics of that one though.

As far as restricted calibers for hunting, one shooting buddy had a 338 lapua magnum licenced for both target and hunting at one stage. But not licenced solely for hunting.
As usual the postcode lottery applies so you may get lucky, or you may not.

Over on the continent I've shot a 9.3mm double rifle which is pretty popular with hunters for boar and large/dangerous game.
However it is a honking big round, about 1000 joules higher energy than 30-06, and of course has a chunky recoil to boot.
And for even the biggest deer here it is a bit overkill, in terms of energy, cost, availability and shoulder beating 😁

I've shot a good few Drillings. The trad German hunting gun,mostly in 16 GA which is or was the trad continental shotgun caliber.Basically a SXS with a large caliber under slung rifle barrel in anything from 6 to 9.3 X whatever,as there is a bunch of obscure and common German calibers for them.

They are expensive and complex enough mechanically,guns due mostly to setting up the POI of both shotgun and rifle barrels which are usually set to a 50 meter zero on the left shotgun barrel with shotgun slugs and rifle barrel.

Add a sub caliber device in a 22 or 6 mm caliber and its quite a versatile gun for European hunting. Most practical one for Irish hunting I've ever seen was ironically owned by a German here in Ireland years ago in the old Irish Shooters Digest magazine.It was a so called "Wald Laufer"[forest walker] in 22lr and 12 GA.But had the 22lr barrel built into the shotgun rib.These are pretty rare, even in Germany.

I use a 22 magnum/20 Ga Savage which as built sometime pre -1964,as there are no official factory serial numbers on it.Apart from its barn door latch trigger,it is a surprisingly accurate gun with both rifle and shotgun using slugs almost both shoot on point.Which seems to be a thing with the older savage guns,that started to fall away as the design went thru changes over the years,until today where it is appaling apprently.

Do the rolling blocks come in deer legal cartridges like .30-30? And can they only be chambered for rimmed cartridges?

Maybe not in rolling block,but they do I belive in falling block,which is proably THE strongest action you can get.Ruger I think makes one in their Nr 1 falling block line.

The 5,6x57 is nearly dead in this country. There were a good few about when a 22 centre fire was all that was allowed for deer stalking. It’s,IIRC, actually 228 using a 74 or 78 grain bullet. I know a couple of people using the 6.5x57r in combination guns, 16 bore sitting over a rifle barrel. 7x57, 6 people I know, myself included use this caliber either as 275 or 7x57. One guy I know used a 30-30 Marlin lever action for a while. It was a bit worn so he jacked it in.