#1
I have a Dearmond Pilot with both a split coil and single coil pickup. I've always preferred the split coil because it has a lot more presence and balls. I like the single coil on the bridge as well, just not quite as much. That being said, I've heard people say that a split coil pickup would get buried in a mix with heavy guitars in the mid to low register for genres like death metal, thrash metal, and doom metal. Is that the case usually, or is it simply a matter of properly EQing the amp? Would the bridge single coil work better for genres like that?
#3
The simple answer is yes, it would work well. P-basses are justly valued for cutting through a mix. It's not down to whether it is split coil or not but to the tonal balance you get with that particular pickup plus it's positioning along the strings. 

Having said that getting a mix right is about all the instruments in the mix. For live work you all need to eq and set your volumes with the rest of the band in mind. If the guitars are boosting their bass frequencies and you are pumping out sound at the same frequencies they are going to merge and sound muddy. To cut through you need more mids and top. The bridge pup will potentially help here but blending the two pups can create a mid suck out which you want to avoid and the bridge pup on its own will often lack body. A really punchy sounding speaker or eq'ing the amp is a better friend in this situation I find. In the PA panning guitars hard left or right helps separate them from the bass as can tweaking their eq. In the studio the same things apply but you have time to experiment with settings and can work on each part of each song to get the mix right. Live you are often just playing a set with an initial mix unless you have the luxury of a decvent engineer.
#4
Mixes have a lot more to do with separate instruments clearing space for each other than it does with pickups. Perhaps the biggest PIA is the guy who sounds heavy in his bedroom and wants to keep his sound that way in a band situation.