If some professional has a video, and they want to do a voiceover for it, how do they do it?
Do they first remove unwanted clips from video and then record audio using audacity while seeing the video as it is running?
If you are referring to tutorials or commentary then yes, you would record the original performance (perhaps with computer sound), while thinking of your audience. When I record tutorials for software, I often perform the example while talking, then go back and edit it down to a reasonable length. I remove unwanted sound, then re-voice the finished edit from a script (doesn't have to be fully written, can be bullet points).
This way I can make mistakes in delivery and restart as many times as I like.
Then i lay that recording back over the edited pictures. At this point I may find that I need to make the video longer to accommodate the new voice over. So there is a little bit of give and take. It is quite hard to manage picture lock without sound.
The process you are describing is called ADR (automated dialog replacement) or dubbing - after picture is locked, lines are recorded separately in a controlled environment and mixed to match the appropriate sonic perspective the required. If you were to adr all of the lines in a vid, you would also then add in a room tone/ambience to give it a soundbed to live in. If you are replacing a few lines that cannot be salvaged, you would then mix in to match the remaining lines and mix the adr'd lines within the ambience of that scene. Most hollywood and large productions are 75-85% ADR - shooting conditions/locations or props etc create a ton of sonic pollution, so most on location sound recordists do their best to capture what will be the reference track once the project goes to post production.
There are a few common recording techniques - productions that go through the trouble of ADR are usually a higher production value than Audacity, most likely using ProTools or other industry standard DAWs.
The two basic ways to record ADR would be to loop the video/original lines to be replaced in your recording program, so the talent can watch and deliver multiple takes to match the on screen performance. Another method, is to have the talent parrot their performance; you start recording, the actor hears the performance, and then just mimmics right after. I find the second easier because youre not asking someone to anticipate the video they want to match, they can just focus on trying to mimmic their own earlier performance - let the recording engineer focus on if the audio syncs with picture, let the actor focus on delivery.