1. The Istari came via ship (though not all together) to the Grey Havens where they met Cirdan, when they arrived they informed him of their divine mission to counter Sauron (and here Cirdan secretly gave Gandalf Narya in the belief that he, and not Saruman, was the wisest of the order) and so they clearly did not arrive with amnesia as to who they were, what their purpose was and where they came from and on whose behalf.
2. Yes they had to learn stuff anew, but you are highlighting the wrong section of the quote. They didn't need to learn anew their magic or connection to Iluvatar or The Utmost West. What they had to learn anew was the experiences of being truly incarnated or "embodied" in a physical form with wants, needs, emotions they ad angelic beings were not used to as made clear with this other section of the Unfinished Tales chapter on the Istari you are quoting:
"For they must be mighty, peers of Sauron, but must forgo might, and clothe themselves in flesh so as to treat on equality and win the trust of Elves and Men. But this would imperil them, dimming their wisdom and knowledge, and confusing them with fears, cares, and weariness coming from the flesh."
3. Yes their memories of Valinor were dimmed, but this was done to prevent their pangs of homesickness for the divine perfection of Valinor to distract them from their mission in ME, made clear by the above quote you use "....though they knew whence they came the memory of the Blessed Realm was to them a vision from afar off, for which (so long as they remained true to their mission) they yearned exceedingly. Thus by enduring of free will the pangs of exile and the deceits of Sauron they might redress the evils of that time." This quote clearly shows, through describing the "exceeding" nature of their "yearning" for home, with yearning already being a word with strong emotional emphasis, the strength of the memory of Valinor on these beings even as dimmed as it is, with the further compounding language describing the "pangs" of "exile" supporting this. The suggestion being that unless dimmed and made hazy the memory of Valinor would distract the Istari from their important mission. What is clear is that the memory is merely dimmed, not removed, as that would also be detrimental to their ability to counter Sauron.
4. Tom Bombadil, a merry soul he may be, and deeply ancient. But he is not so far as we can tell angelic in nature nor does he come from Valinor. Nor is he listed as one of the few who came to know the true nature and origin of the Istari (that list being the White Council - which Bombadil was not ever part of) So there would be no way for him to teach Gandalf the ways of the Ainur or his status as a Servant of The Secret Fire/Iluvatar. Not to mention that even if Tom was in such a position it would be completely against his character to actually do this.
Apologies for the long ass post
Let my inner literature lecturer loose for a second there!
