My hope here is to emancipate you from your uncertainty, however unfounded (yet reasonable) that uncertainty may be. You see, the fitness-instruction racket can be laden with complex and sometimes overly precise formulas for scheduling exercise so much so that it can leave people feeling unable to do even a push-up without first discussing it with the authorities.
But to answer your question about doing push-ups and crunches each and every day: Go for it. The only time you ever need to skip a day before you can do the same exercise is when the weight is so great that you have created the conditions for overload and muscular hypertrophy. The body needs 36 hours to recover from that sort of exertion. Lifting the weight any sooner wouldn’t hurt you but it also wouldn’t help you—it would likely only delay the benefit of the earlier workout.
If you possess a basic level of strength, however, a push-up doesn’t really fall into that category. Up until 20 or reps, a push-up is more of a muscle-toning than muscle-building exercise. And after 20 reps it becomes primarily a test of local muscle endurance (which may be one reason why the results of old-school push-up tests used by the military and the government tend to not correlate with overall fitness statistics).
Likewise, the general thinking on crunches and sit-ups is that one can do them every day of the week, not just because prisoners do so with such great outcomes, but because bodyweight core strengthening is more along the lines of flexibility training than strength trainingit simply requires less recovery. If you feel you need time to recover, then take a day off. Follow your ...er ...gut on that one.