The person who hands in the mission gets the choice on whether to share the mission rewards (which includes standings, ISK and LP) or keep them for themselves. You can also have it remember your preference, so if you’ve already set this option and want to change it, you need to go to ‘reset settings’ in the escape menu to change it.
Yes, highest standings between agent corp and faction takes precedence UNLESS any of them are under -2. In that case you can only access level 1 agents.
Take a look at the ‘Faction standing repair plan’ for some more ideas on repairing standings.