1. strings are not really THAT big of a determining factor on "lowness" bass, amp, and how you EQ them both will determine that a lot more. If you'd like to learn more just start browsing (and "searching") the STRINGS forum, that's what it's there for man.
2. a "J" bass is a Fender Jazz style bass, a "P" bass is a Fender Precision style bass... they are the "standard" most popular styles of basses. The main differences are the thickness/width of the neck (P neck's are bigger), the shape of the bodies (J bodies are more asymetrical), and the pickups (P's have... well, a P pickup, and
Js have... you guessed it, a J pickup). The tone is different on each for probably all of these reasons. A J bass is arguably more versatile, while a P has a fairly specific (and sought after) sound to it.
3. Try turning the amp up with having anything plugged into it, if it hums it's the amp. Then turn it up with just the cord you're using plugged into it (making sure the other end is in the air and not touching anything) if it hums then but not when the amp is by itself then it's probably the cord. Then turn it up with the bass plugged in... I think you get the idea.