First of all. This is a perfectly good place to ask your questions. You could also as on "The Dark Siders Den." There are actually several places here on the CAS City forums that are also just fine. Most of us really like trying to provide help for New Folks. Or even Old Folks as it twer.
For Myself, I don't know if I can be of real current help as I retired from being an active Gunsmith 7+ years ago, so some of my information may nit be strictly ..... current. But here goes anyway.
Most if not all conversion cylinders made for a .36 Cal Percussion gun were bored for a 38 Special or shorter (Kirst) cartridge. The Kirst was intended to limit cartridge over all length (OAL) to require the use of 38 (358) Hollow Base Wadcutters or similar Hollow Base projectiles that would expand to work in a .36 Bore of .375. I am not currently familiar with anyone loading Heal Base ammo commercially. Don't quite me on that. I haven't spent any time chasing that. The Kirst cylinder is actually too short to load standard 38 Special ammo. it's deliberate.
R&D or Howell conversion cylinders are long enough and chambered for 38 Special. Shoot (almost) any commercially loaded 38 Special ammo. The problem is most commercially loaded ammunition is loaded with 358 diameter bullets which exhibit spotty accuracy and leading in a .375 barrel. 358 bullets just kind of rattle through the bore. However, usually you can still hit cowboy targets a 7 yards. All you need is Minute of Buick at 7 yards. Any better accuracy will REQUIRE Hollow Base Wadcutters or other similar Hollow Base bullets.
Shooting other than Hollow Base or Heal Base bullets will not harm you or your gun. You may find however, you can't hit anything smaller than a Buick. For heal based bullets of the correct diameter, you will need to check and see if your cylinder is through bored. If the cylinder has "throats" to align 358 bullets, 375 - 380 bullets will avail you naught. The throat will swage the bullet down to .358. Your only financially viable route is to reload. Finding commercial Hollow Base Wadcutters is a real groan. A real expensive groan. You may also find a severe fouling problem with a Conversion Cylinder.
I personally had terrible experience with .38 conversion cylinders in my .36 Navy Pietta's. With BP and Subs, the guns would foul out in about 4 rounds. Since I don't shoot Smokeless, I sold the cylinders off. I wish you good luck with yours.
Oh, almost forgot. Short answers:
Yes. Your suppose to shoot Heeled or Hollow base bullets to match up with the bore.
Not for safety reasons. It's so you can hit something. Accuracy.
Yes. If commercial ammunition is not specifically labeled as Hollow Base or Heeled, it isn't.
Buffalo Arms is it. Far as I know. (Don't quote me).
Reload.