OK, so I have a player who is in his 3rd arbitration right now. Today is arb day, so I can no longer sign him to a long-term contract.

If I arb the player, and then trade him to another team, what are the odds that he will sign a new long-term deal with the new team? Will he automatically go FA with them because *I* arb'd him 3 times, or will the new team still get a fair shake at re-sign him long-term?

You can try to sign him to a LT contract today.

When you're on the "Arbitrate Player" page for the player, there's a box of text in the middle of the page.  In the middle of that text there's a link to "Sign long term", or something like that.  That brings you to the page where you can offer the LT contract.

I don't have any answers to your other questions.

40 seasons of HBD under my belt, and I had no idea that link existed. Thanks!     

I've extended a guy who was arbed three times by his former team and then traded to me.  This is the guy.  I'm thinking that his extremely high patience and/or makeup ratings might have something to do with it.