It is certainly not customary, but I don’t believe it would be outside the rules to use a rope halter for showmanship. Double check your rule book to be sure.
The one area where a rope halter could be dinged points, is that you are likely not going to get the “snug fit” that is preferred for showmanship where your halter needs to be adjusted correctly to your horse’s head. Now, if you don’t really care how you place and are doing it for the experience, then it won’t matter.
Yes, this is common. Usually if you enter the ranch horse classes, you cannot enter western pleasure or horsemanship or show trail. It depends on the show, depends on the classes offered, etc. Always read the rules!
When I go to a ranch horse versatility show where they have Ranch Reining, then I also cannot enter the regular Reining. It’s against the rules. Which is fine. They are trying to keep the Ranch movements different, as they should be. But if a show does not offer Ranch Reining, then I could do (for example) Ranch Riding and still enter the regular reining. (because Ranch Reining wasn’t an option at the show and I wasn’t entered)