I think only the more aggressive tribes and the ones less willing to assimilate will be wiped out completely. The five civilized tribes and ones east of the Mississippi will probably live settle lives for most part and be very integrated into American culture but be treated as second class but still treated much better then black people. They could end up created a less binary Jim Crow segregation system in the south later on. Something like Hispanic countries had with the cast system. They will probably survive in the south in noticeable numbers especially in Appalachia due to less immigration and raw population while in the north immigration will probably cause them to become nearly unnoticeable. Out west is where I see them possibly becoming like Roma except for New Mexico and Arizona where they can become more like the assimilated and settled Native Americans you see in Mexico. They could drift around as farm laborers in the plains?
Not in Appalachia, they were already pushed out of there. What the Five Civilised Tribes had was some nice land in the South. They had elites who had some manner of status (not quite as good as whites, but still respected by white outsiders) and also the poor who resented those elites (who were more of your stereotypical Indian). The elites owned slaves and were often mixed-race, and were prosperous. These people would assimilate into Southern white society, creating some mixed-race people of all classes and no doubt a situation where some would identify as "descended from Indian chiefs" but also as "totally white". If enough Southern elite are descended from American Indians like this (this would take an earlier TL), then we could have a nice addition to a Southern national myth where the "Southron" race is descended from Anglos, Celts, and the surviving "mound builders" (since those who built Cahokia, the Pinson Mounds, etc. are far too advanced to be the same "primitives" their ancestors fought--this pseudohistory was common in 19th century America) who compose the leadership of the Indian race and despite the fall of their civilisation still rule over the "lesser" Indians. Said "lesser" Indians, who would still exist, would be blended with poor whites and perhaps some blacks, and would be largely ignored as long as they remained peaceful.
I don't think they'd much be like the Roma, simply because all Indians had different lifestyles than how the Roma have lived. They were hunters, gatherers, and simple agriculturalists, not quite the lifestyle of the Roma. If their experience with the horse and later Comanche experience with cattle is anything to go by (in addition to Navajo experience with sheep, etc.), they could transition to pastoralism, which in turn breeds some problems. Some Indians were farmers, and some like the Five Civilised Tribes did very well with slavery and plantation agriculture (like Cherokee Stand Watie, CSA general). OTL mixed-race individuals were cowboys, trappers, and other such professions in the West, but by the late 19th century mixed white-Indian people were not treated nearly as well as they were a few decades before then (although they still existed of course, and were treated better than blacks).
In the North and probably Midwest the American Indians will mostly go extinct. They'll blend with locals and be forgotten. You'd have more cases like the modern Pequot, where they get "resurrected" by locals who perhaps have distant ancestry in that group (there's a lot of controversy with them, including issues which no other than Donald Trump himself brought up back in the 90s). On the other hand, if "American Indian" as a racial category is no different than "African American", I can't see the controversies involving gambling rights, etc. coming about.