When prestiging you get 10% of the prestige you add when you battle a gym. If you add 3600 prestige to a gym, that's 360 exp. You don't get additional exp for placing a pokemon.
Whether or not you get 1100 exp for clearing a full level 10 gym (11000 prestige), I can't confirm at this point. Perhaps it's a 5% gain only (550 exp).
Edit: https://pokemongo.gamepress.gg/gyms
A full clear of a level 10 gym should yield 1050 exp. 100 exp per opponent and 50 exp for the clear bonus.
