I do just a few push ups (10 to 20), not an excessive amount. It helps me to increase focus on my work during a day.
Is a warm up necessary before doing these or can I go without?
Generally, if you want to perform an exercise set then a warm up is preferred. We all know this, but I found some great documentation on what a warm up actually does, and why one should do it. Most notably:
What does a warm up do?
As for the type of warm ups you could do, one comment to your question suggested jumping jacks and that's a great warm up that will accomplish the goals above. Another one is burpees, which is kind of encroaching on the push up kinetics a bit, but still worth a mention. Mountain Climbers as well, just to name a few.
Note: I didn't used to be a warm up person (1/4-1/2 mile jog before weightlifting was about it), but I've since re-thought the whole warm up idea.
Unless you are getting at over 50% of your max there is no reason to warm up.
I definitely do not support jogging or any other form of cardio before lifting. You want your muscles to have tension when working out.
What I do suggest to a person doing push-ups in the morning. Get down slowly and do the first 5-10 slowly with perfect form.
Yes, do "warm up".
There are many ways you could build up a short routine that's safe end effective. Of course, doing nothing else than your 20 push-ups is a lot better than doing nothing, but it may be far from optimal. You may think that the push up is not a difficult exercise, but you still put your bodyweight on your wrists and shoulders. So "warming up" for you specially would mean to gently move the muscles without straining, and not "increasing your pulse to 70% of maximum".
Here's an example of what I would do:
This works for me but it's probably best if you experiment and build up your own routine which works for you.