Push-ups are great exercises! As long as they’re performed properly they can have a tremendous amount of carry-over to other movements and they can somewhat easily be loaded. Take a look at this video and watch me demonstrate some poorly performed push-ups with my elbows flaring, immediately followed by well performed repetitions at both a flat level and a decline level.
If you just want to see the movement performed then fast-forward to about 3:15!

Definitely one of the most beneficial exercises you can perform! I like to superset the push-up with the bench press. In other words, after I complete a set on the bench, I immediately hit the floor for some push-ups until I can’t do any more!
Absolutely, also a great way to create hypertrophy, while supporting the important internal/external rotational roles of the shoulder that proper bench pressing inhibits!