First off, I never learned to slap, and in my present situation I have no need of it. Tom Petty's music doesn't have any slap in it! Second, the reason I have a cover over my pickup is because I feel that the true P-Bass sound is somewhere between the pickup and the neck, so the cover forces me to play there. Coupled with well broken in flatwound strings you get that strong thumpy sound associated with classic P-Bass sounds of the past. It's really just a personal preference, since the P is such a versatile instrument used for many different styles.
As far as Paul goes he started with the Beatles by playing guitar, and when he switched to bass he played it with a pick. That's just the way he prefers to hold his pick, which I think is a large triangular pick. Myself, when I play with a pick I use a Fender 351 medium cellulose pick, held with just my thumb and index finger, and my other fingers are flayed wide open. Been playing like this since the 70s, so it's what I'm used to. Once again, personal preference.