Is our Lord’s time over at the end of Shabbat?

And the fact that Rabbi Tam wrote that the sunset is much longer than we measure here in Israel, do you think he verified this against the sunset in northern France (where he lived) or did he rule this out regardless of the reality around him?

I have no idea.

In Israel, too, it takes time for the sky to completely darken.

Rabbeinu Tam's entire method is based on incorrect astronomy.
He claims that after the ”first” sunset” (the usual sunset) when the sun is hidden from our eyes, it has not yet left the world but passes through the ”thickness of the sky” and comes out of a window and out, and this time takes another hour, and this is what is called the ”second” sunset.
These are things that began in the Gemara, and you can see the controversy there already.
It is clear that the sky has no thickness and this whole thing is a bluff. Indeed, several poskim today have written that there is nothing to fear from Rabbeinu Tam's method because of this. Many are stricter than Rabbeinu Tam following the Shulchan Aruch (there is also disagreement about his opinion, one can clarify both systems from him, although it is quite clear that he follows the same path as Rabbeinu Tam in both Beit Yosef and the Shulchan A.) (There are also Briskerim and Hasidim who strictly follow Rabbeinu Tam even at the beginning of Shabbat and perform rituals until the time of the R.T.)