Most tangs are rent-a-tang in a 120G, but if you get into it with the fish's best interest in mind and are willing to pass them on to a good home (even if it means giving them away for free), then I think that you can still responsibly own one for a while. I am going to leave tangs like valmings/unicorn out of this since very few could ever have room for them and you cannot usually give those away when they start to grow.
Most of the Bristletooths (as mentioned above) purple and mimic tangs can probably live their whole lives in a 120G.
Tangs (and most fish) are kind of like the idiots on Big Brother. If you meet all of their needs with food and good environment quality, they will generally get along (not always). If you starve them, or poke-the-fire, then they will act like asses. Tangs need a lot of food and even though they are mostly reef safe (not completely), they aren't really reef friendly when they get big because you need to feed them a lot. If you have them in a FOWLR with high nitrate and phosphate levels, they can start to get HLLE and get mad. Hippos need vitamin E, so you might have to supplement. If you use a good pellet, like New Life Spectrum, then you have a good chance of raising some nice fish. I used to feed mine NLS Pellets, Formula 2 Pellets, Nori and Mysis/Octopus/Squid. Tangs are omnivores, not herbivores.
I have had hand-me-down tangs that ate coral, so they aren't completely reef safe. I had a purple that ate zoas, salifin that ate a guy's prized acans, yellows that ate zoas and leathers and a sohal that got a taste for rose 'nems. Not sure if the sohal actually ate the roses, but it did enough damage to kill some of them before the guy caught it. Except for the purple, the other tangs were pretty good sized, like 6+ inches and probably not being fed enough anymore. You should be feeding enough that they are growing.
While you QT these fish, get them eating out of a fish trap - take the door off, and put the food in it every day and have them go in to eat. Get them eating pellets too. This can really make it easier later. Remember that hypo will not get rid of all disease, so plan your QT regimen accordingly.