yes, as leader of the largest party. please remember you dont vote for the leader, but for MPs. the PM is the MP who can most likely command the House of Commons. (actually not sure it even has to be an MP, we've had Lords as PM in the past)
Not since 1963 when Lord Home was appointed to succeed Harold McMillan and had to renounce his peerage and stand in the Commons. He was only able to do that after Wedgie had successfully campaigned to drop his Viscountcy.
I would guess however Dave would leave the Commons in 2020 and go upstairs. Interesting that no mention of repealing the 5-year fixed Parliamemtary term though, a lot of people think that was a mistake as a permanent measure.