This [answer](http://mathoverflow.net/questions/12688/nonsingular-normal-schemes/12689#12689) 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](https://edocs.uis.edu/mmaji2/www/Research/Dissertation.pdf), 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](http://www.ams.org/bull/1974-80-06/S0002-9904-1974-13612-8/)). A paper by [Lyubeznik](http://www.jstor.org/pss/2946619), especially Section 11, has many such results, even for positive characteristic cases. It also includes many relevant references.