This answer of mine briefly discusses Hartshorne conjecture and some related questions about smooth subvarieties of $\mathbb P^n$ of small codimensions. It links to Hartshorne's original paper, which I think is still the best source to answer your questions 1) and 2).
As for 3), you can also look at Zolbani's thesis, which has a lot more details then his research statements mentioned by Steven.
(That's all I know, I would be very interested in what's new about Hartshorne's conjecture as well).
EDIT: Today while answering another question I was reminded of a line of research which can be viewed as evidence for Hartshorne's conjecture: smooth subvarieties of small codimension behave cohomologically like complete intersections (this was discussed in Section 2 of Hartshorne original paper). A paper by Lyubeznik, especially Section 11, has many such results, even for positive characteristic cases. It also includes many relevant references.