Why does Frodo offer the Ring to the Galadriel when Gandalf specifically told him NOT to offer it to anyone!? And why did Frodo follow her from his camp out? Plus, he didn't even look like he was scared or even surprised when her hair blowing away and her talking in a scary voice and her turning all evil mode?! Was he just tired? Is Frodo just immune to being scared or surprised or was he just tired?
This isn't obvious from the movie, but looking at the book gives us some clues for approaching it.
Why does Frodo offer the Ring to Galadriel?
Short version: He's starting to realize how big of a deal this quest is going to be, and he doesn't think he's up to it. This is clear from the book:
'You are wise and fearless and fair, Lady Galadriel,' said Frodo. 'I will give you the One Ring, if you ask for it. It is too great a matter for me.'
Fellowship of the Ring Book 2 Chapter 7: "The Mirror of Galadriel"
Even in the movie, Frodo's offering of the Ring to Galadriel comes immediately after she gives him a pretty pessimistic view of the outcome:
Galadriel (V.O.): The fellowship is breaking. It has already begun. He will try to take the Ring. You know of whom I speak. One by one, it will destroy them all.
Remember that all Frodo knows of "adventuring" comes from Bilbo's stories; Bilbo may have faced danger, but he never had the fate of the entire world on his shoulders. Plus, his old friend Gandalf had just died saving the rest of them from certain death at the hands of thousands of Orcs and a demonic fire-monster.
Frodo is in way over his head, and he knows it. He sees Galadriel, who looks very competent and powerful, and figures "surely she can do a better job of this than I can." The fact that he is hilariously wrong in this assumption is the point of the scene.
Why does Frodo follow Galadriel in the first place?
This is a movie invention; in the book, Galadriel explicitly motions for him (and Sam) to follow her:
[']I hope very much that before we leave we shall see the Lady of the Elves again.'
Even as [Frodo] spoke, they saw, as if she came in answer to their words, the Lady Galadriel approaching. Tall and white and fair she walked beneath the trees. She spoke no word, but beckoned to them.
Fellowship of the Ring Book 2 Chapter 7: "The Mirror of Galadriel"
This obviously doesn't happen in the movie, but the script hints that some magical power is driving him forward:
The Fellowship are asleep on their beds. Gimli is snoring loudly. CLOSE ON: BARE FEET tread soundlessly across the lawn. CLOSE ON: Frodo's eyes flicker open...as if by instinct. GALADRIEL, her White dress glowing in the moonlight, glances at him. Frodo follows her...as if drawn by an invisible force.
What this "invisible force" could be is unclear. Possibilities include:
Galadriel's own magic. She demonstrated earlier that she could communicate telepathically; it's not a stretch to suggest that she could also communicate non-verbal thoughts like "I want you to come with me." Note that she's not remotely surprised to see Frodo has followed her.
The Ring is driving him. Being claimed by Galadriel is everything the Ring wants; she's a powerful individual who would try to use it directly against Sauron, be destroyed, and then Ring would have achieved its ultimate goal of being reunited with Sauron. Leading its current bearer into a one-on-one situation with Galadriel, particularly if it can sense his growing doubts, is a great way to do this
The will of Eru. Unlikely, but distinctly possible. Even if it's not Eru intervening directly, it could be the same sort of Providence that moved Frodo to accept the burden of the Ring in the first place
Why doesn't he react when Galadriel has her "moment"?
The book doesn't discuss Frodo's reaction to Galadriel's sudden display of power (which is much less pronounced than the movie anyway), but the script suggests that Frodo does react:
Galadriel: You offer it to me freely...I do not deny that my heart has greatly desired this.
Galadriel suddenly seems to rise in stature before Frodo's eyes. Frodo is suddenly afraid of her.
Re-watching the scene from the movie, Frodo looks pretty freaked out basically from the moment he stops looking in the mirror. It's notable, though, that when Galadriel is doing her thing, he does get noticeably more frightened-looking: his jaw drops, and he takes a couple of steps backwards. And once she's calmed down, he has this very concerned look on his face and looks at the ground as if to say "okay, maybe this was a bad idea."
It's very subtle, but the cues are there.
There are several questions here, some of which are specific to the movie. All movie-specific questions are best answered with 'because peter jackson'.
In the book, Frodo accepts Galadriel's offer of counsel, and so of course he follows her to the mirror. He sees Galadriel as a source of help. And when she 'becomes threatening', he is not freaked out, because she does not, in fact, threaten at all. She is making a point. To paraphrase: "Frodo, I shall not accept the ring because it would turn me into a monster."
I recommend a close comparison of Galadriel and Tom Bombadil; two cases in which Frodo looks to unburden himself of the ring to a force greater than himself, which hands it right back. Metaphorically, the ring is the problem of 'the children of Eru' -- human/hobbit-ry can't pass it off to someone else, and neither, as the practical instance of the situation, can Frodo. Passing it off at very best kicks the can down the road, or at worst trigger a worse disaster.
How come Frodo offered up the Ring to the Galadriel when Gandalf specifically told him NOT to offer it to anyone!?
Because Gandalf is not the boss of Frodo, and Frodo has his own choice in how he decides things. Gandalf is not his boss, but is Frodo's wise and trusted friend. Galadriel—one of the oldest living things, sentient or otherwise—is also wise, and Frodo trusts her. Therefore, he seeks her help in unburdening himself of the ring.
And why did Frodo follow her from his camp out?
Because she is wise, and he trusts her.
Plus, he didn't even look like he was scared or even surprised when her hair blowing away and her talking in a scary voice and her turning all evil mode?!
Yup, he didn't even.
Was he just tired?
Naw... it's probably just that she's one of the wisest things alive and he trusts her.
Is Frodo just immune to being scared or surprised or was he just tired?
I would say it has more to do with Galadriel being one of the wisest living beings and that Frodo trusts her.
Frodo, as he repeatedly says throughout the books (and films) does not want to carry out the quest to destroy the ring, mostly because of fear. He offers it to Galadriel for the same reason he offers it to Gandalf- he knows they are wiser and more powerful than he is and wants them to do the quest. Frodo, before going on the journey to destroy the ring, wants to go on an adventure, but doesn't really want to endure the danger- but he is brave enough to carry it out anyway. He is like Christ at the Garden of Gethsemone- he doesn't want to do it, because he knows he'll suffer and likely die, but he is willing to do it because he knows there is a purpose larger than himself which he is serving- to prevent Sauron from enslaving Middle-Earth
Frodo offers the Ring to Galadriel because his quest is a suicide mission with approximately a 0.0000001 chance of success. Frodo is naturallry afraid of death and torture for himself and his companions.
Frod is also humble enough to know that it is impossible for him to resist the temptation of the Ring long enough to destroy it, and hopes that the wise Galadriel might be stronger than him.
It is a literal miracle that Frodo is able to resist the Ring temptation long enough to reach the Crack of Doom. Considering all the minutes the quest lasted (approximately 129,600 in 90 days) the odds against Frodo naturally succumbing in that one minute are very great. Frodo lasts as long as he does because Eru grants him extra strength to resist the temptation as a reward for his patient suffering.
Frodo offers the Ring to Galadriel because that is what anyone would do in his situation.
Except that almost nobody else would be dutiful enough to offer to destroy The Ring in the first place, or unambitious enough to be trusted with it by the council in the second place. So almost nobody else could ever have been in his situation.