So I want to make an audio/video material, where among other things I will sing part of a welll known copyrighted song. I may sell the final product. I want to know if I break copyright by singing some verses of the song ?

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As you described, yes.

Yes, of course you do. You can get a license to cover the song.

If you release it and make money you should pay royalties to the song writer

Yes, this is considered copyright infringement. One of the exclusive rights of a copyright holder (meaning that they are the only person legally able to do this) is to create derivative works based off of their copyrighted material. Derivative works would include a sequel to a novel, a play based on a movie...and a cover version of a copyrighted song. It is the copyright owner's right to determine who can create derivative works, so you'll need to ask for permission before you do this. Good luck!

You need to get a licence to cover the song in question, and pay publishing royalties to the author(s). Once you do that, you'll be fine.