I'm not sure about this, but I believe that the late '90's MIM Standard Jazz bass came with two pickups of the same length, unlike the original and later '60's and Geddy Lee (as well as Highway One and Road-worn series) Jazz basses. These came with pickups of different lengths, to allow for wider string spacing nearer the bridge. In other word, the Fralins may not fit. It might be a good idea to measure the pickups that are in there to make sure.
Anyway, you'd probably be better off with some new strings and a good set-up, and an amp that you can play through, before trying out new pickups.
For flats I would recommend Thomastic-Infeld, for rounds there are a bunch of choices, the Daddario's are fine to start out with. You have to
compare tones to figure out what you like, as well as tension, string feel and other details.
You can't really hear it without an amp. I don't know if there is a GK combo that you can afford new, but an old 2X10 400 should be in your price range. When I saw jazz (music) I started to think SWR Basic Black, which was discontinued in 1999, I think, but then when you say rock (and country) I started thinking about an older GK, which can be very versatile.
You could also just spring for a Hartke kick-back, or maybe even an amp-cab rig (used) in your price range. Or an older Ampeg 1X15, maybe. But you really do have to compare and contrast not only tone, but size, weight, power, features, etc. - and then make a decision. If you find something you think you might like in your price range - go for it. Then you'll at least have a basis for comparison. You'll want to find out the "truth" about power ratings and efficiency and sensitivity, but for now just get the best amp you can.
As far as tubes in bass amps, there are many with a tube used in the pre-amp, but it's not a vital part of the equation like with a guitar amp. An all-tube bass amp is really heavy and really expensive. But some people love 'em (
).
Good luck with your endeavors.