Should I have a days rest when doing pushups?

I would advise that you do take some rest days as that will help your muscles grow more and actually become stronger. As with the punching bag you are working many of the same muscles as with the pushups so when you take a rest day with the pushups you should also not use the punching bag on that day. Though with that being said there are other things you could do on your rest day, such as walking or jogging, or really any other lower body exercise as that does not work the muscles you are trying to rest, I know this may sound complicated but I hope you get what I mean.