In my workout, I have 2 days specifically for weighted bodyweight exercises (pushups, squats, and chinups). One day on Friday is light like 3x12 or 3x8. On Monday which is my heavy day, I do 5x5 and sometimes go up to 7x2 on some weeks. I don't do every set to failure, like 7x2 is maybe done with my 4-5 rep max. Is doing that much weight and that little reps bad for me?

No it's good for your max strength, which will be beneficial for your ability to develop your explosive strength.

Just make sure you get lots of other work in, and don't overeat. This way you'll get stronger without gaining weight.

Ok, thanks. I'm actually eating less because I need to lose some weight. At 200 right now, I need to get to maybe 170-180. Then hopefully gradually I can gain some muscle and get into 200+ being very lean though.

Mostly you only need strength in the last 6" of the punch. You need speed for the first 1' or so in order to get to the target in a timely way. I'm not sure what the fixation with push-up is, but realistically one only needs to do short push-ups at the end of the arms to develop punching power muscle-strength-wise.

Ok, but as long as doing full pushups 7x2 with heavy weight isn't going to be bad, I'll continue to do them fully.