OK--I'll take a stab at answering your questions.
1. The bore diameter is more than likely as shown, .44. The easiest to load would be a .430 ball.
2. The cap size is probably #11. Most modern cap locks use that size.
3. For the ramrod, it's most likely 3/8. Go to a hobby shop and get some short lengths of wooden rods, in 5/16, 3/8, and 7/16. Try them out, and then buy the proper hickory rod from a place like Track of the Wolf. Most of them come with a brass fitting on one end, and most are threaded in 10-32 (for the jag you'll need for loading and cleaning). You should also have a muzzle protector, which is a short collar that prevents the dirty ramrod from abrading the barrel, a nipple pick, and maybe a ball puller "just in case." It's best to have both ends of the rod with the brass fitting, pinned to make sure it doesn't pull off inside the barrel.
4. The correct loading procedure is simple--pour a measured amount of powder down the barrel (NEVER from the horn), followed by a lubricated patch and the right size ball. There are myriad formulae for the lube--some use spit like I do (I put a pre-cut patch in my mouth after working up a good amount of saliva, while I measure and pour the powder), but Lehigh Valley Lube, Mr. Flintlock--probably the same formula--and Hoppe's #9 Plus work well. You would benefit from using a "short starter," which would have a short brass cup for initial starting, and a longer (8" or so) rod for ramming the patched ball down a bit. Then, use your full-length ramrod to push the ball all the way down, in short increments to avoid breaking the rod and jamming it into the palm of your hand. Make sure the patched ball is well seated on the powder, and it also helps to make marks on the ramrod, showing when it is properly loaded and when it is unloaded. That helps to avoid the dreaded "dry ball." Any space between the patched ball and the powder will be dangerous. I have a handle on the end of my range rod, and give it a firm pat to make sure the ball is firmly seated. I see people slamming the rod down numerous times, but that probably deforms the ball.
5. Get one of the graduated powder measures, which will show amounts in 5-grain increments. For a 44, you might start with 50 grains. Using a bench rest, shoot at least 3 rounds, and see how tight your group is. If it's pretty wide, and you haven't been flinching, pour in 55 grains, then 60 and so on. The groups should tighten up, and when they start to get loose again, you'll know the best load. The variables are ball size, patch thickness and lube. Questions, questions, questions. Try 10 oz bull denim, available at most fabric stores, like JoAnne's. I've tried a lot of different fabrics, and the 10oz bull denim works best for me. It has to be 100% cotton or linen. I pre-cut my patches, although some folks cut the material in strips, holding them in their mouth like I do the pre-cuts. Pre-cut and pre-lubed makes it simple, but spit is free. For a 44, 1 inch square will probably work OK.
6. BP substitute? Not advisable, unless you can't get the real thing where you live. If you're shooting black powder, use black powder. There are lots of posts on this forum about how the synthetic powders ruin barrels, are hard to clean, and are slow to ignite--they don't work well in a flintlock, for instance. BP comes in grades, and the most common are 2f and 3f. In a short-barreled gun, 3f might work best. It creates higher pressures than 2f, which works best in long barreled, large bore guns.
7. Cleaning--here again, there are as many opinions as there are lubes and patch material. Most say that plain cold water works as well as anything, but only for real BP. Most use alternate wet and dry patches, and whatever you use, keep pushing them in and out until they're totally clean. If you use water, you probably should follow with WD40, to disperse the water. Then use a rust-inhibiting oil, like RIG. Check the barrel with clean patches in a couple of days, looking for rust, which can form very quickly. I check my guns every week, even if I haven't been shooting. You can also use solvents, like Track's or "Muzzle Magic." Then, you won't need the WD40, but you should still use the rust inhibiters. I don't know what your gun is like, but many cap locks have a "drum" with the nipple screwed into it. If the drum has a clean-out screw, you can push pipe cleaners though to make sure it's clean, and some people dissemble their guns, put the breech end in a bucket of water, and flush the water in and out. That cleans the drum and nipple, but be sure to use the WD40, and store the gun for a while muzzle-side-down so everything drains out.
I'm sure that if I've forgotten anything, other members will chime in, for this forum is full of very knowledgeable and helpful folks. Good luck and welcome to the wonderful world of black powder!
8. Safety rules are largely the same at most ranges, but BP clubs usually insist that you load at the bench, and don't prime until you're at the firing line, with nobody standing in front of you. Keep your barrel pointed up or downrange. If you join a club, they'll fill you in, and watch you to make sure you are proficient. Some public ranges have BP areas. If you shoot at a range with the rest of the "suppository" shooters, expect some looky-lous and questions. You might convince someone to look into a BP gun for themselves!
