Do people use screen recording? If they do, will that include sound?
Definitely not in a professional setting. If you're making an amateur production, and you only need a few seconds or so, a screen capture would be fine to cut a bunch of work time. If the vid is only a few minutes long, downloading it is probably just as fast. Many screen capture programs can record system sound, so you're covered there.
Do they actually download whole youtube videos to disk just to get a quote or something on 2 seconds included in their project?
In a professional production, yes, every time. Always work from the closest you can get to the source file, despite file size or download times. That's just a pain of the industry. There's several good tools available, but done of them are spammy and on the ad/malware side of things. Use your best judgement before you install anything. YouTube-DL is good, but it's a CLI, so you might find the learning curve frustrating. Some videos can be download via YouTube settings, but it's an opt-in option, so it's rarely available. I don't have this need pretty much ever, so I'm not up to speed on the best way to rip internet videos.
Do they just input an url and select a start and an end and then have it automatically added to the timeline (being able to drag start and stop timestamps afterwards)?
I don't know any video editing programs that can do this, but I bet there's a few. However, consider what the program had to do behind the scenes. It downloads the videos anyway, so this would only be a slight convenience over downloading yourself.
I'll reiterate the other answer's legal concerns. Every jurisdiction I know has a "fair use" law for exactly things like this. Short quotes to build on your own work is understood virtually everywhere as legal. My rule of thumb has always been 30 seconds or less.