The P has a deeper sound than a Jazz, though the Jazz is more versatile and has a friendlier neck to boot. Anyway, I started on and still play guitar too, and I don’t have big hands, but I find a P neck to be absolutely fine (and yet, whenever I play a Jazz, I’m always reminded that the Jazz neck is even finer, lol. But still, the P neck isn’t any big deal. Like, I play two short scales and two Precisions, they’re all fine — sold my Jazzes, actually). But get whatever bass appeals to you; I’m just saying that I wouldn’t avoid the P because of its neck.
For good quality at an affordable price in a known brand, you might try a used Made In Mexico Fender. Assess them pretty much like buying a used guitar. Meanwhile, Ibanez is a different kind of standard bass brand that has some lower priced models you might consider.
If you’ll be a bedroom bassist or mainly a recordist as a bass player, a short scale would be best, I’d think. They play basically the same as any other bass and sound great while being arguably easier to play, and easier on the body. However, if you’re planning on playing out on bass in a hard rock/metal band, you’d probably want to consider the look (regardless of scale length, actually). But a short scale makes for a fun, easy second instrument for a guitarist to have around. Relatively compact, too, so less banging into into stuff with the headstock (Ooof!) before you’re used to the extra length you’re wielding.
About pedals, I have 12 pedals on my [frankly, unnecessarily large] bass pedalboard, and of those pedals, only 3 are dedicated bass pedals: an MXR Bass Fuzz Deluxe, an MXR DynaComp Bass, and an Electro Harmonix Bass Clone (chorus). Another pedal on there, a Catalinbread Dirty Little Secret MKIII, which simulates a Marshall amp, has an internal switch to cop a Marshall Super Bass bass amp sound, which I have switched on, so it’s a pedal half-dedicated to bass. And my other distortion, the Catalinbread SFT which simulates an Ampeg amp, is probably intended equally for guitar and bass.
Also, a Mu-Tron Micro Tron III envelope filter I have on the pedalboard is another pedal that is equally for guitar or bass, or any other electronic instrument, really.
Of the remaining 6, two, a tuner and an ABY switch, are "utility pedals" that can be used with essentially any electronic instrument. Finally, four are sold as "guitar pedals". So, as others have said, feel more than free to experiment with your guitar pedals and see what works (some will, some won’t)