It hasn't happened yet for any NFL team to play the Super Bowl at their home field (though two teams played in the game when it was held in their cities/areas).
If it happens, all the arrangements will remain unchanged. Both teams will be treated as "away" teams when it comes to their uniforms.
The "home" team would probably feel more comfortable than their rivals using locker room at their stadium (though they could draw the visitors locker room), and being close to their training facilities.
Would such a game give the "host" team an advantage? Well, probably only mentally, because all the other aspects would remain just the same as they would be if they'd be playing the game anywhere else.
The ticket would be split evenly, so this wouldn't make much difference, though the local fans could try some ways of getting more places for themselves.
The home team gets to choose which jersey they wear (white or not-white) in every NFL game (not just the Super Bowl, but also the Super Bowl) - see the NFL rulebook:
Home clubs shall choose their jersey color (either white or official team color), and visiting clubs must wear the opposite.
As such if a team was nominally "Away" but was at their home field, the other team would be able to choose which color uniform they each wore.
There's nothing in the rulebook indicating locker rooms, and so it's unclear what would happen there; my guess would be they'd be allowed to use their normal locker room, but it would probably be up to the commissioner to decide ultimately, if the other team preferred to use the home locker room.
As far as whether they'd get a real advantage - it's unknown, and can't be known, since this hasn't happened, and home team advantage itself is very hard to determine as it is.