Good luck on the Birthright (you'll need a Constitutional amendment).
Nope. The practice of birthright citizenship is entirely a convention. Trump could end it tomorrow with an executive order.
No, the President doesn't have the power to unilaterally overturn the Constitution and 200 years of SCOTUS precedence.
It's the law, not a "convention".
Nope. The relevant part of the Fourteenth Amendment reads "All persons born in the United States, and subject to the jurisdiction thereof, are citizens of the...etc. "
Under the jurisdiction thereof goes back to English Common Law and generally means something like "who can compel you to go to war". The United States government can compel me to fight, but not a Canadian who is on this side of the border shopping. He isn't under the jurisdiction, even if he gives birth.
The question has not been decided repeatedly. There are some dicta cases, or whatever that's called, but everything hangs on the Wong Kim Ark case from back around the time of the Chinese Exclusion Acts. First off, it was a monumentally stupid ruling, the dissent blows it out of the water, but secondly the case was not typical for what we see today. In that case Wong Kim Ark's parents immigrated legally to the United States and made the US their permanent home. Their kid, Kim Ark, was born here, but then, as a young man, decided he wanted to return to the motherland. When that didn't work out, he moved back to the US, but in the time that he was gone, the Chinese Exclusion Acts were passed and so he was prevented from entering. He sued, and eventually the Supreme Court ruled in his favor, holding that his parents were under the jurisdiction of the United States when they had their kid and, therefore, he was a citizen.
That's the ruling you claim gives illegal aliens, birth tourists, and god knows what all automatic citizenship.