There is no difference between home sessions and "official" sessions. Games played at home and properly reported are just as official as those played at stores or conventions. One of the main points of how the organized play system works is so that characters can go jump into playing scenarios no matter where they are, be they at home, game store, or convention.
So yes, the same character can be played in any type of organized play scenarios, mixed and matched, on the same chronicle sheet/set of sheets.
The only limit is if you solo then you only get credit for one character not the number you play as far as OP is concered.
So for example you can solo a whole party of four at home but when you do OP with others you only get to use at most one of these charaters.