Hi Paul

paul_h

“Watch me doing five push-ups.”

One comment I would make is that it would be unusual to add the exact number of push-ups in your first sentence. That's because the use of the -ing form tends to focus on ongoing activity rather than on complete act(s). So, if you know you are going to do exactly five push-ups and you want someone to watch the whole set of 5 complete acts, then you would usually use the infinitive: 
- Watch me do five push-ups.

If you said something like "Hey! Look at that guy doing push-ups." I would interpret that to be a reference to an activity (push-ups) which was already in progress: "Look at that guy (who is) doing push-ups".

When you use the -ing form of the verb, the focus is on the activity, and it often is not a complete activity.
When you use the infinitive, there is more focus on the complete act, the act as a whole.

In addition, some verbs are followed only by infinitives, and some are followed only by gerunds. You just have to learn those.

There are other things you need to keep in mind when it comes to using the -ing form or an infinitive, but I hope I've answered your question at least partially.