Think of your standard pellet as one of those badminton birdy things. When you hit it, it takes a few feet and then it stabilizes itself and starts flying straight. That's exactly what a diabolo pellet does! They technically dont need any spin, like qball said.. but we put a spin on them to get them straight in a shorter amount of time which improves accuracy. The diabolo pellet hasnt changed any in several decades, but better rifling and barrels improved the accuracy.
Slugs are the opposite, they need spin to stabilize them the whole way and often lots of it! So slug barrels are often faster twist rate, which pellets dont typically mind. You'll find the heavier pellets like more spin, and the lighter pellets like less. I know guys do good with the really heavy 25 cal pellets in slug barrels.
Twist rate depends on the manufacturer. I think LW barrels are available in several twist rates last I looked.
Wobbly or spiraling pellets are damaged pellets, caused by dirty grooves that cut an uneven pattern on the pellet or too tight of a barrel that damaged the head. They are thrown off balance aerodynamically and are spiraling as they try and straighten out.