you could add a row - seated cable or bent-over BB, etc - think of your joints in pairs of opposite motions - and shoulders move in several planes : Mil Press+Lat pull down, BenchPress+some Row motion
If you are doing heavy BP - you might want to add some light "pre"hab exercises for your rotator cuff - better now then as rehab

"feeling the burn" is not related to quality of workout or intensity. don't worry about it.

rest btw sets depends on: 1) intensity of the effort and 2) what your training goals are - are you increasing intensity by decreasing rest for example? There is one training program where (for example) you take a 3RM weight, do 3 sets of 2, gradually decreasing rest btw sets over a period of weeks until you are doing 6 reps of your previous 3RM.

if you were doing sets of 3-4 reps at 95% of 1RM you might rest 2-3 mins between sets so that you are recovered enough to do another intense effort. On the otherhand, if you are trying to increase the intensity of the workout, you might decrease rest btw sets.

Probably 60-90 secs would be typical for most types of lifting, or use supersets w/out rest (so you rest as long as it takes to set up the other exercise, do the set and return) or work with a partner - and rest for as long as it takes to reset the weights and the partner to complete a set.