Rest between sets if fine, I actually encourage it. You want to leverage rest so you can do the most work in the least time. If you can squeeze out 30 pull ups in a workout rather than 20 because you rested an extra 3 minutes total that is a big win. So if you can do level 6 rather than level 4 by addin a little rest by all means DO IT!
Do not go to failure, stay 2 or 3 under failure, break up your sets.
Do lots of small sets 2-3 under failure and you will see you numbers go up much faster.