Who's the eldest being in Middle Earth, Tom Bombadil or Treebeard?
Bombadil was "eldest and fatherless".
Treebeard was merely the eldest Ent. Note that the Ents were awakened to sentience by the Elves, so Galadriel (who was among the Firstborn of the Elves) was older than Treebeard as well.
Note that the age of any particular Elf is problematic in general - very few of the Firstborn were mentioned by name, which is not the same as "only a few of the Firstborn were still alive in Middle Earth (much less in Valinor)".
Is "Sauron" (lit. "abominable") a name that he despises and does not permit his underlings to speak, and if so, why does he have his messenger refer to him as "Lord Sauron the Great" and a servant refer to himself as "the mouth of Sauron"?
Because they weren't speaking English, and it's annoying to have names for characters vary within a literary work? Or possibly because the comment about "he permits his name to be neither spelt nor spoken" was an exaggeration by Gandalf (the wisest of the Maiar) when speaking of Sauron (the most powerful of the Maiar)?
Why does Gandalf warn people against using devices "of an art deeper than we possess ourselves" when talking about the palantir and yet have no problem with with the fellowship using all sorts of magical items of arts deeper than they possess (glowing elvish swords, daggers from the barrow, the Phial of Galadriel, Galadriel's box of earth, etc)?
Because he meant "an art deeper than *I* possess" when he said that (he was using the Royal "we"...). The work of Feanor was beyond even the Maiar, unlike sharp pointy things made by Elves and handed out by same to diverse characters....
So Galadriel knows Sauron's thoughts that concern the elves, but didn't know of Saruman's betrayal, or never saw relevant to mention it to Gandalf?
Saruman was NOT an elf (it's not clear what he was - perhaps a Maiar like Gandalf). So "thoughts that concern the elves" didn't apply. Note that the thoughts in question weren't "thoughts of interest to elves", but "thought about the elves".
Oh, and the mithril shirt pretty much had to be hard enough to stop a blade, or there was no point to it. Most likely the "supple as linen" was marketing-speak for "amazingly light and flexible by the standards of a mailshirt". What, you didn't think they had marketing in Middle Earth? Marketing was one of those evils even older than Sauron that was mentioned in passing....