Is it true that when Jack went to work for DC in 1970, he asked them to assign him their worst-selling book and that's how he got Jimmy Olsen?  And while you're at it, how come he didn't just go back to Challengers of the Unknown?

When Jack went to work for DC then, he didn't particularly want to do any existing comic.  He actually didn't want to do conventional format comics at all.  He wanted to experiment with new sizes, such as what we would now call graphic novels but he didn't have much choice.  He also didn't want to do what he called "someone else's characters" but DC insisted that he take over at least one monthly title that they were already publishing.

Jack was told he could take his pick of "any book in the place," although I've always assumed that if he'd picked certain titles, like if he had a sudden urge to take over Batman, they would have quickly talked him out of them.  In any case, he looked over everything DC was publishing and didn't see anything he especially liked or felt was suited for him.  So he said to Carmine Infantino, who was running the company, "You pick something for me.  Give me whatever you like and I'll do it.  Give me your worst-selling book if you like and I'll make it your best-selling title."  But he also told Infantino not to remove someone else from a job they already had.  Jack, being a Depression-era kid, hated the idea of anyone being fired.  So he asked for a book that was currently without a regular writer and artist.

At the time, Superman's Pal, Jimmy Olsen more or less fit that description.  Its editor, Mort Weisinger, was leaving the firm and another editor, Murray Boltinoff, was about to take it over.  Boltinoff had not formally selected his creative team yet.  (It would probably have gone to writer Leo Dorfman and artist George Tuska.)   Jimmy Olsen was not DC's worst-selling title by any means, though its sales were then on a steady downward trend.

Also, there had been some talk of Jack doing something with Superman and coming up with some ideas to reinvigorate that character's comics.  Since Superman appeared in Jimmy Olsen, that seemed like a likely place for Jack to work out some of his ideas about the Man of Steel.  There were also some folks at the DC office who wanted to see Kirby revive the Newsboy Legion and since James Olsen was something of a newsboy, that seemed like yet another reason to assign Kirby to the book.  So Jack, mustering as much enthusiasm as he could for a comic he didn't especially want to do, took over Jimmy Olsen.

As for Challengers of the Unknown…at the time of Jack's return to DC, that book was marked for cancellation.  It was, in fact, featuring reprints of Kirby's original issues from the fifties.  As far as I know, there was no talk of him returning to it.  DC had pretty much written it off — it might actually have been their worst-selling title.  The company had already given up on it and starting Jack off on a book with such a poor recent sales history would have been a needless handicap.

What happened to Jack's work on Jimmy Olsen?  Who redrew his work and why?

As I explained in this article, DC executives decided his drawings of Jimmy Olsen and Superman didn't look right.

Why can't they have better inkers "re-ink" the Kirby comics that were inked by bad inkers?

There are two aspects of that question…one technical, one ethical.  The technical one is that you can't "re-ink" an inked comic.  That is, if Jack pencilled a story and then Fred Badinker inked it, that's probably all that exists of that story.  If you then give those old pages to Phil Goodinker and tell him to "re-ink," all he can do is trace what Badinker inked and smooth things out and kind of guess what Kirby did that was now lost.  That's not inking Kirby.  That's creating a new piece of art that reinterprets how Badinker inked Kirby.

There are a few cases where Xeroxes or stats exist of Jack's pencil art, and those could theoretically be used to have Goodinker do a tracing and "reinking."  But even tracing good stats that way isn't the same as actually inking Kirby pencils, and there are also the moral considerations to consider.

Jack hated the idea of someone tampering with his work after the fact to "improve it."  As far as he was concerned, when a story was done — even if it was inked by Vince Colletta — it was done and should not be touched again.  (Jack was bothered a lot less by poor inking than some of his fans are.)  I didn't like what Colletta did to the early Fourth World comics but even if you could magically expunge his contribution and make the books look like they would have if Joe Sinnott or Mike Royer had inked them, Jack would not have wanted that.

As noted in the article linked in the previous answer, before DC reprinted Jack's Jimmy Olsen issues, there was a brief discussion of having someone — probably Steve Rude — retouch the work to be more "Kirby."  Jack's friends and estate were willing to entertain the notion because it was not so much a matter of reworking Kirby as of changing one set of revisions for another, more sympathetic set.  But it was finally decided that it might get even farther from what Jack intended and that since there was a question, it was better to err on the side of not fiddling with completed work.