Challenges? Eh, I'd take the most I guess. Weak challengers are good for getting a feel of how your team works and just plain gathering exp and strong ones allow me to improve on my strategy and prediction skills as well as pinpoint any serious flaws in the team (Speaking of NB experience here, but the same would pretty much apply if it were for real. ^^). Arrogant challengers? Oh, I'd definitely take those on, particularly if it's obvious that they're all talk. Bringing the reall puffed up ones down in a 6 0 kind of way is just too satisfying. xD The times I wouldn't battle would be...
1. If it's blatantly obvious that I couldn't measure up to my opponent and the situation didn't force me to battle.
2. If my pokémon weren't in the shape to battle. (Although I'd have to make an exception in forced situations like emergencies and suchlike)
3. If I have something more important to do or an appointment to keep.
4. If my challenger had some other agenda than just plain battling me.
5. If the battleground was rigged in my opponent's favour, specifically against me, or a combination of the two. The exception of forced situations applies here too.
6. If there were ridiculously high stakes involved in the battle. (Again, I would have to make an exception for the times when the battle was forced, say...if my opponent had staked something that I desperately needed to get)
7. If my challenger was a pathetic whiner who had spent the last two weeks challenging me again and again just because he/she lost against me before.
If it were an IRL thing, I guess that virtually no experienced/famous trainer would have the time for every challenge being sent their way, so yish. I generally try to meet the demand, though. ^^