Wow, difficult questions! It's all so subjective! I'm sure others will chip in soon, but for my part, I'd say playability of the SC is far superior to any J-style bass I've tried, due to the short scale (and hence lower string tension) and narrow fretboard. Js feel big and clunky to me in comparison. But as I said, it's subjective: some people prefer a longer scale, a higher tension and wider spacing, and until you've tried an SC in person, you don't know which camp you're in!
Neck dive is usually noticeable in SCs, but adapt your playing position accordingly (I find I subconsciously use my right forearm/elbow to brace the bottom of the bass so the neck doesn't dive) and it ceases to be an 'issue'.
Soundwise, again, I can't speak for everyone, but to my ears, an Alembic always sounds like an Alembic. I've never heard one do a convincing impression of a Jazz. People tend to buy Alembics because they like that distinctive Alembic sound - it can be mellow, harsh, cutting, soft, punchy or pillow-like, but it always has a quality that it distinctively 'Alembic' and therefore very distinct from a Jazz. But the sound is versatile, and can certainly be used in all the genres you list (I use my SC in various bands playing the whole range of styles).
The best suggestion I can give is to have a listen to some recordings featuring a short scale Alembic: I'd recommend Paul Simon's concert in Central Park, featuring Armand Sabal-Lecco on Alembic bass for much of the set (he switches to a Jazz, I think, for a couple of the older numbers). None of the original (i.e. non-live) recordings of the songs in this live set featured an Alembic, so Armand sounds very different to the original bass players, but that's not to say his Alembic doesn't sound 'right' in that context. Just different. And again, in many cases and IMO, I actually prefer Armand's sound on these live versions to the recorded originals.
If you're dead set on the 'Jazz tone' for the songs you're playing, you may find that an Alembic shorty isn't for you. But if you're prepared to try something different, but which can work equally as well, then you may find that you like the Alembic difference!