If you're using BIAB, use F5 to turn on and off parts. If you're using RB or another DAW, just use volume envelopes or mute where you don't want them to play.

As for timing, it's very important to learn to play in time, that's a given.

However, there are songs out there where the tempo changes as the song goes along. For example the chorus may be sped up a little bit from the verses. You'll have to learn to listen for that in your covers and adjust your tempos accordingly. That is one possibility that could be causing it to not seem to feel right like the songs you hear on the radio when you're playing to a steady beat.

The way you learn parts to put in your production is by listening preferably in good headphones so you can hear all the parts in songs you want to emulate - what instrument, when did it come in, where was it panned. Then it's trial and error in your own productions.

Hope that helps you some.