Lord Voldemort had cursed DADA job, and I was wondering, would renaming the position or opening a new position instead of DADA break the jinx?
I believe that there's no real explanation for this thing. We can only assume about the details.
Though presumably, Dumbledore realized that this curse existed after several teachers came and went. Well, if you're a headmaster of a school and you've been on the job for years, I think you'll indeed notice if such a hiring problem exists, and with one job at that.
And with Dumbledore being Dumbledore, we can assume that he eventually realized that it's highly likely that Tom Riddle was the one who placed the curse. As presumably, the curse appeared after he was refused the DADA job when he asked for it.
"You see, we have not been able to keep a Defence Against the Dark Arts professor for more than a year since I refused the post to Lord Voldemort." —Albus Dumbledore
We don't know if Dumbledore already did this renaming thing. Though it's kinda obvious that Dumbledore never did figure out how Tom did the curse. Or else, he would have broken it already.
This jinx was cast by Tom Marvolo Riddle, who was by the time known exclusively as Lord Voldemort, to curse the Hogwarts school's Defence Against the Dark Arts class to ensure that none of its applicants could hold their post for more than one year. It is unknown what the requirements and incantation of this jinx are, or if it could be used for other school subjects or other establishments and careers. The jinx will be lifted once the caster is dead.
And then again, scratch what I previously stated. If it occurred to Dumbledore that the nature of the jinx/curse was such that it'll be lifted once the caster was dead, it may be that he intentionally didn't remove the curse at all. But instead continued to observe what would happen, as we know he had doubts about the reality of Voldemort's death (the first time he died), and his doubts were indeed proven to have credence. It's possible that the continued existence of the curse was one of the hints to him that Voldemort is indeed alive, so Dumbledore had something like a Is-Voldemort-Alive alarm that triggered every year in the form of the DADA job jinx.
It's possible that he's had wishful thinking too, that a professor would finally stay for an entire school year. Which would signify that Voldemort is finally gone for good.
No. In the last book, when Voldemort took over Hogwarts, the job was replaced with "the dark arts".
"used to be Defense Against the Dark Arts, except now it's just the Dark Arts." harrypotter.wikia.com/wiki/Dark_Arts_(class)
Even after renaming it, the teachers still only lasted a year (because battle of Hogwarts!)
For the second one, most likely yes. Remember, Tom Riddle cursed the job after being refused the job. He came to Dumbledore to become DADA teacher, then was turned down. He then cursed the job, while inside Hogwarts. Also, think about all the protective enchantments protecting Hogwarts. There is no way to curse something inside the school from outside the school.
As for the third one, I've got nothing.