I'm not entirely sure on the state of the US when it comes to this stuff so I can't really say on if Clint can push for a group home for Josh, although he has alluded to having the power to do it through various meltdowns over the years by Josh.
Clint is likely the name on the apartment, like it was in the previous place, but its paid through Josh's SSI. All other expenses like food, Internet and so on are all paid by Josh. Josh paying his own Internet was a big thing when he moved out from the basement. Josh receives nothing from Clint unless he has a very deep negative number on his bank account, but has the apartment in his name to control things on the Josh front. Mostly dealing with problems and having proper contact with the apartment management so that he can get through to Josh since no one else in the world has that power whatsoever.
Regardless of what happens though, even if Josh lost all his income from wand sales, while Josh is still getting SSI he can still get apartments if he gets kicked out as there will always be a shithole in Casper that cobra can afford through it, and there is no real harm on the bank of Clint for this. However it would be a different issue if something happened and Josh manages to lose that SSI, or do something fucking stupid.