-ing or not?

All of them are correct.

What are you doing right now?
I am working on the website.

What do you do when you are having fun?
I work on a new website. (I have fun building websites.)

Oh I see... so the -ing refers to what happens right now, is this correct?

And what about the second example, could you say "they are protesting" when they have been protesting all week, but they don't do it right now, maybe on sunday?

geoyoAnd what about the second example, could you say "they are protesting" when they have been protesting all week, but they don't do it right now, maybe on sunday?

Yes. I read about the protests in Egypt, but I don't understand why.
They have been protesting for two years, and continue every Friday after services.

I can ask at any time:

AlpheccaStarsWhy do they protest? (OK, but used less that the continuous.)

So, in analogy, when someone asks: "What are you doing right now?"

Could you also say: "I work on the new website"?

Would it sound strange/not natural?

geoyoWould it sound strange/not natural?

Yes, because if the question is in a continuous tense, the answer is usually in the continuous tense. The exception is if the continuous is used to ask about something in the future.

What do you do for a living?
I work in the post office.

What are you doing now?
I am sweeping the floor.

Thank you, that was very helpful