In general, no. If you want to apply to YC, just apply to YC. Going through some other program first is a negative--the advice is usually bad, there's extra dilution, etc.
We ask about when they've done the impossible. You also get a pretty good feel for this after asking someone hard questions for 10 minutes. Not pushing back at all is bad, and pushing back too hard is also bad.
We randomize everything first, and then look through each morning to see if some interviewees need domain expertise, and then swap those around as best we can.
Unfortunately we don't do that--it just doesn't fit our model. Plenty of companies have done YC at a later stage and still thought we increased their value far more than the 7% we take, though.
I think startups are not well-suited for open-ended R+D unfortunately (hoping to do something about this). The best time to move from research to a startup is when you've developed technology and have a plan to make it into a product people will want.
Totally depends on the startups. In general it's good to monetize early, but there are very important exceptions. This is the sort of thing we have to give individualized advice about to every YC company.
We might do something like rolling admissions, but the advantage of the batch model are huge. It's so helpful to the startups to be around other startups at roughly the same stage.
(dang please don't ban me for a low-quality comment :) i couldn't resist but will not make it a habit!)