Regarding the question about why would Galadriel let Sauron enter Eregion if she was the ruler, I guess it depends on what version of Galadriel we accept.
I have always found it quite telling that in the Lord of the Rings, Elrond wishes that even the 3 had never been made, but Galadriel only wishes that the One Ring had not been made or at least never been found.
Sauron found their weak point in suggesting that, helping one another, they could make Western Middle-earth as beautiful as Valinor. It was really a veiled attack on the gods, an incitement to try and make a separate independent paradise. Gilgalad repulsed all such overtures, as also did Elrond. But at Eregion great work began – and the Elves came their nearest to falling to 'magic' and machinery. With the aid of Sauron's lore they made Rings of Power ('power' is an ominous and sinister word in all these tales, except as applied to the gods).-Letter 131
This is dangerous close to what Galadriel is suggest to Celebrimbor in the Unfinished Tales, if we accept Celebrimbor as the maker.
'What wrong did the golden house of Finarfin do that I should ask the pardon of the Valar, or be content with an isle in the sea whose native home was Aman the Blessed. Here I am mightier.'
'What would you do then' said Celebrimbor
'I should have trees and grass about me that do not die-here in a land that is mine,' she answered.
It seems to me that Galadriel may have rejected Sauron, but she was still drawn to his offer. Her gift was in reading people's intentions. She may have seen that Sauron was hiding his malice, but none the less she desired what he had to offer.
Again she alone of the Wise wished for the One Ring to fall into her possession. Perhaps Galadriel allowed Sauron into her land to try and gain knowledge from him, but keeping him under close watch at the same time.
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