By far the easiest and best conversion is the gated conversion offered by Kirst. It does require cutting a loading port in the blast shield, but once done the gun can be loaded and unloaded without removing the cylinder. I've used these on both Remington and Colt pattern guns, and they are excellent.
These two have Kirst gated conversions.
There are a few very good gunsmiths out there that also do conversions, and of course if you get
really ambitious you can do your own...
This 'Bulldogged' Remington has a bespoke 5-shot cylinder and breech ring that includes a rebounding firing pin. It's chambered in .45 Colt. The cylinder was turned from half-hard 4140 steel and line-bored. For the Walker conversion below I bored-through the existing cylinder and made a breech-ring.
The .44-55 Walker cartridge uses 55gr. of FFFg under a 200gr. heel-base .451 bullet. This may seem pretty powerful, but it's actually lighter than the load these guns were designed for.
As to ammo for a cap-and-ball gun that has been converted- Kirst recommends using lead bullets loaded to less than 1000 fps. Factory 'cowboy' loads fall well under this power level. If you buy a factory-built conversion, such as the ones offered by Cimarron and Taylor's,where the gun comes complete and already converted from the factory, you can safely fire any standard-pressure factory ammo of the correct caliber through them. These guns have passed European Proofing, so modern ammo isn't going to phase them.