I think it is common to get a few fruit on first year grafted tree, maybe a coupe years for significant fruit. I think they respond very well to pruning and Mulberry fruit on new wood so you can prune severely and still have fruit. I have known people to even pollard them. Many cultivars are very vigorous and almost require it if you want fruit within reach as opposed to “sheet n shake” .
“Does alba= white, rubra= red nigra = black ?” Yes but not in reference to fruit color. IE: many Alba are red and black fruited. If I recall the color denotation is in reference to small hair like fuzz on the buds or young growth, but I am not sure about that. It does not denote fruit color. There is quite a bit of dispute on how many species there are and identification, and that some of the species hybridize further muddies identification.
Nigra does have pretty much have acceptance as the best flavored as a species. More Complex than just sweet to me, but many others are not far off. I like most all I have tried with the exception of some Alba being just plain bland, other Alba can be great. I don’t know I ever had a Rubra or a Nigra that was not at least pleasant.
If you really want to try pot, some of the Nigra certainly trend more bush then tree. My Black Beauty is taking a bush form, but Nigra is a supposedly more finicky in climate, and probably will not like a humid climate. I have a Geraldi Dwarf, which I have seen a couple other close spellings as Gerardi as well as different species listing, but think most classify it as Alba. It is a true dwarf with very close fruit nodes and stated to stay very small and have to think it would be a good pot candidate.