In a melee if one of my opponents has used the redirect from the Crown Regent. If I use Myrcella's ability to switch titles with them do I get a redirect with the Crown Regent?

I would say no. The once per turn is a restriction of the title. Its not tied to the player, but to the title, so regardless of who has it, it still can only be used once.

I agree with Josh here. Similarly, if someone uses the Hand of the King's influence, and THEN it is taken with this ability, it would not be available for use.

However, if you stole the Master of Coin after someone used it to count income, that would allow you to count +2 income.

I don't agree with this reasoning.

We're playing a GJ-on-GJ match. Let's say you have Saltwife out. You use her ability (limit once per phase) in the Challenge phase. We know this ability is "card specific" because you could certainly use the ability on your second copy of Saltwife, right? Now let's say that I take control of the one you used already with Seductive Promise. Can't I use the Saltwife I took from you to save one of my Ironborn characters, even though you have used her once already this phase? Similarly, what if you use your Sniper Drogon in the challenge phase, then I take control of him with Reek. Can't I use his ability once this phase, even though you have already used him once?

Or how about this scenario: You use the Limited Response on your Castellan of the Rock in the Marshalling phase. Then I take control of him in the Challenge phase. Does that stop me from using my Icy Catapult later in the round? If the restriction follows the character without reagards to the player, shouldn't it?

When you look at situations like that, it is pretty clear that restrictions and limits on cards are tied to the player as well as the card. I don't see any reason why shifting the title from one player to another would prevent the new holder of the title from using its "once per round" effect just because the old holder of the title had used it - assuming the new holder can do so legally.

Or, if you don't like the "you can see from other examples" reasoning for the limit following the player, not the card, I'd offer this argument:

The rules for the Crown Regent title say, "This title also has a special ability that allows you to redirect one challenge each round." Since "you" always refers to the controller of the card, "you" have not redirected a challenge this round when the title changes hands. So the new holder should be able to do a redirect as well.

Wow, I actually didn't know that about cards like Salt Wife and whatnot. Huh. Yeah if that's the case, then you can clearly use the redirect. Touche there Ktom. Learning something new everyday I guess.

The Limited Response exemple seems inapplicable to me: the restriction is on the player (though it's also printed on every card with a Limited Response), not the response itself (unlike Salt Wife or Make an Example). But considering Make an Example (limit 1 per phase), it would be weird to prevent one player from playing one just because another player played one earlier in the phase.

Anyway, you can still give someone a useless title, like master of laws after draw or master of coin after the player has counted income.