Yes, I've done that loads of times First, you need a Youtube downloader - I use this one http://youtubedownload.altervista.org/ You download the video using that program and save it on your PC as a flv file. You then use the same program to convert it to an MP3 file and cut out the video bits Once you have an MP3 file you can send that to your phone via USB, but if you want to cut a bit out of it you need a program like Audacity http://audacity.sourceforge.net/ which allows you to mess about with the file, cut bits out etc.
I use Zamzar, copy and paste the Youtube URL link into it and it downloads and sends to your E-mail. Re-download it from there and jobs a good 'un . Takes about 5 - 10 mins. Got to make sure you change it to a MP3 though.