The half cheek is traditionally the bit used for showing purposes. I have one full cheek snaffle but don't care so much for it for driving--it is nice for training in some cases, because it's less likely to pull through the mouth, but even on my riding horses I rarely used a full cheek unless I was riding a horse that didn't like the bit folding & dropping down in his mouth--then I would use a full cheek with the keepers on, and that way the mouthpiece doesn't drop down in the center. As far as I'm concerned the full cheek serves little purpose unless I want to use the keepers.
It is true--the single jointed bit may dig into the horse's palate--a problem especially likely in Miniatures with their small, shallow mouths. The double jointed bit doesn't have that same action. The double jointed bit won't give the same nutcracker action as the single jointed one either.
Copper mouthpieces are fine--I have several copper bits, ordinary jointed snaffle and french link--as long as your horse doesn't grind on the bit at all. If a horse has any inclination to pull the bit into his teeth and chew on it, a copper mouthpiece gets chewed up very quickly. So, I don't use the copper mouth bits until I know the horse isn't going to chew it up--I use the stainless steel bits most often--my SS french link bits have the copper link and really I find that is enough copper.
One thing that I really dislike about so many mini bits--the have bulges on the bars of the mouthpiece. Some people don't seem to care, a few even think those bulges are a good thing, but I simply will not buy a bit that has those bulges. IMO that is a sign of a poorly made bit and it is not what I want. When you use one of those bits, when you use the lines all pressure is centered on those bulges. So, instead of an even distribution of pressure the horse feels pressure under those two little spots. That is not what I want.