Im curious about the P bass pickup, what are some of the qualities of the P setup. I want to do a Jazz bass with the P/j pickups

pros/cons?

The P bass pickup lends a smoother output with better low end capability but still gets some nice mids. Being two pickups wired in series it has a higher output than a J pickup and sense the coils of each pickup portion are wired opposite polarity it cancels out most of the noise associated with a J single coil pickup.
By contrast a J pickup has a little more grind if you will and more pronounced Highs. A J bass setup when both pickups are at or near full volume exibit some of the qualities of a P pickup in that they become more noise cancelling as the Neck pickup is wired out of phase with the Bridge Pickup in similar fashion to the P pups.
A lot of people like the J setup. I on the otherhand prefer the P bass tone over the J's. I use a Seymour Duncan classic stack (noise cancelling) as they get 90 percent of the tone of a J single coil but little of the hum and buzz associated with a J by itself.
On my P/J bass I have a Seymour Duncan SCPB-3 Quarter Pounder in the P bass position. Clean but hotter output than the typical P pickup. I use a Seymour Duncan Classic Stack J in the Bridge position. I put a mini toggle to enable/disable the J Bridge pickup and pot so I get a pure P bass tone. I can then add the J pickup as necessary. I get a good full tone (slightly leaning toward Jazz bass sound). You can use a 250K audio taper with a Push/Pull switch to enable the J pickup.

would it be plausible to run jazz control plate with the P/J setup

You can use the standard jazz control plate. One other option is to use 250 Ohm stacked concentric pots so you have volume and tone for each pickup as the pickups behave differently with a tone control. You can buy a jazz control plate with just 3 holes like on the Fender 62 reissue jazz basses. I have done this setup and I used a Fender for the P and a Seymore Duncan STACKED (humcancelling) jazz for the bridge. The separate tone controls will give you alot more tonal options !!

I like the ability to use both P and and a J pickup. I have a P/J style bass myself, however I recommend using a Pot for say the Bridge J pup that has a push/push or even a push/pull switch to totally remove the Bridge pickup and it's associated volume control from the circuit to get a pure P tone. With Mr. Ray's Concentric volume/tone set a SPST mini Toggle for the J pickup enable/disable.
I use a Seymour Duncan SCPB-3 P bass pickup and a Seymour Duncan Classic Stack "noiseless" J bridge pickup.

LTB: Are you talking about SCPB-3 or SPB-3? The SCPB-3 is the single coil, but in post 2 you described the split coils.:confused:

Sorry, the P's are Split Coil Humbuckers. The J can be the SJB-3 or Classic Stack noiseless. I have a 51P RI with the SCPB-3 in it.