Well, tough to tell from those pics -- we need to know what the wires are coming from. But one thing looks odd in the second pic: it looks like you have both the black and white wire from a single pickup going to a single lug on the switch. My guess is that those two wires are coming from your J. and that would explain why you're not getting any sound out of it. But there's a lot odd there --
I think there's a big white, a big red, a b/w pair and a b/r pair. The b/r pair is connected to two lugs on the switch...I don't get that unless one of those lugs is a ground. The b/w pair going to a single lug -- if that's from a pickup, you won't get any sound from that pickup. The white and red -- one should go to the vol and tone, and maybe the other goes to ground? But without seeing all of the wiring, I just can't tell.
So some basics:
Test the pickups by connecting them straight to the jack -- black to sleeve, white to center. Confirm that they're working fine.
Figure out what the lugs on that switch do -- there should be 2 that are switched, 1 that is "on" always, and maybe that 4th is for the shell/ground? Dunno. But for example, you could remove the black wire of the b/w pair and touch it to the other lug with a black and white wire on it -- if you get your J back then I think that lug is connected to ground, and you'll be all set.
The crackling could be a bad pot, could be a bad solder joint. Get the J back first, and then see if the crackling is the same no matter what position the switch is in. In that case I'd look at my solders on the jack. If it's just the P, I'd check its solder joints (I would also test it directly to the jack to make sure it's ok).
BTW: when I do a lot of this back and forth, I use test leads with alligator clips on either end -- then you don't have to keep soldering/desoldering everything.
Let us know what you try and what you find!