If someone beats you and they change forum or PO name, it still counts that they've beaten you. They are the same person, they just have a different screen name.
Ask your opponent before a match starts if they want the battle to count as a Gym Battle, or just a regular one. They can't decide mid battle that they suddenly want to make it a gym battle. It's just not fair. Just be clear from the beginning!
You cannot reject a challenge. Even if you have some personal issue with this person, just battle them! You don't have to get in an argument or even talk to them necessarily. Just battle, count the win or loss, and move on.
If you are going to be busy, please update your Gym or Elite thread so people know they might have to wait a few days before they can battle you.