Push ups are fine to do, heres the proper form
https://www.youtube.com/watch?v=4dF1DOWzf20 just remember you should keep your elbows in not flared out and if you feel main in your joints you should stop.
If you want to exercise with bodyweight exercises heres what I would recommend
https://www.reddit.com/r/bodyweightf...mended_routine
I started this program in October and ive been happy with the results.
As for losing weight, calories in, calories out, simple as that. One simple tip is replacing all drinks with water.