From personal experience and observing my friends especially the ones in ROTC, I noticed that doing a lot of push ups doesn't necessary mean you can bench a lot or bench a lot of reps, but by benching a lot or doing a lot of bench reps will help with increasing your push up amount. Hope this helps.
The straight answer is no, but what really matters is, who cares? I have been doing exercise daily because I was military. Doing bench presses is a vanity exercise anyway, and people just looking for bragging rights, macho stupid stuff. They have no real world applications. Doing sensible body weight exercises is all you need to do, unless you just want to try and show off to your friends.
Believe it or not, you are using different sets of muscles. This can be supported by a Health Instructor and doing both exercises you will end up with a WOW body! Can't wait to see the new you!
Define bench press gains... Are you trying to increase the size of your pectorals or toning them? Push ups is the same as bench press, except it is using body weight, and has a much lower injury risk. If you are trying to increase the size of your pectorals, do 10-15 push ups after each bench press set, but do them slowly. If you are trying to tone your pectorals, do sets of 20-30 nice and fast after you completed you bench press work out.
No. While you are focusing on the pecs and secondarily triceps, they also recruit different muscles. For example, if you can bench press 400lbs, that doesn't mean you can do a push-up if you added weight to your back in order to get added weight + your weight to 400lbs. The converse is also true: just because you can push your body up doesn't also mean that you can bench your body weight. The two are separate things. Another factor is that push-ups are typically done in long sets, or at least that's what you end up striving for. Bench press, and most other lifting, is done with reps anywhere from 1-15 per set (depending on your goals). Do both, as they complement each other.
