Learn English – Using ALL tenses in this text

Grammatically, your sentence

I've caught a cold and I'm staying at home for a couple of days.

is perfect. In American English, you could just as well use the simple past in the first clause. I think in British English the present perfect is expected here.

The simple present I stay at home talks about regular or habitual activities. In your case, you are talking about a single instance of staying at home (even if it lasts a couple of days). You are not saying you stay at home habitually or regularly, but about one instance. So, yes, your use of the present progressive is perfect here.

Functionally, your sentence also gets the message across, by the choice of tenses. And it is helpful, for several reasons, to be able to choose the best way of expressing something.

However, what if you had written

I've caught a cold and I stay at home.

Yes, you would sound like a non-native speaker, but you would still be attempting to communicate in a foreign language, and your reader would probably be able to understand what you mean. So, don't be afraid to make mistakes. Get out there and try to communicate in English. Three fourths of the people who ask questions here just need to go attempt to communicate in English, and learn how to best express themselves by imitating native speakers. They will learn the answers by trying to communicate. And I think that's what you have done, because your instinct was correct. Even though I've said that, don't hesitate to ask good questions on this site. (But, still, learning by actually using the language in meaningful contexts is the best way of learning.)

Your simple past understanding is fine. In your example the present perfect and the present perfect continuous are practically interchangeable. In both cases, David is still working for them. With 'how long' questions it is often the case that the two are interchangeable.

Sometimes there is a slight difference between the meaning of the present perfect simple and present perfect continuous. They both have an effect that is valid in the present, but sometimes in the present perfect the action is complete while in the present perfect continuous it doesn't have to be.

A typical grammar book example is 'I have painted the ceiling' versus 'I have been painting the ceiling'. In the first example, the effect is that I am happy now, in the present; the job is done. In the second, the emphasis is on the process, the action. We do not know if the job is finished, just that the speaker was recently involved in painting the ceiling.

PS I should add that this is British English.