Well, I'm not sure about the muting it part, but I do know that if you download Download helper, for Firefox, you can download movies easily. With the muting part, someone else will have to tell you that, because I can't. Hope that's slightly helpful!
Download off YouTube using Firefox Video DownloadHelper, demux the video stream with FLVExtract. If WMM or WLMM can't read that, use avidemux to change the container to AVI and try again. If it still can't read it, transcode it to WMV using avidemux.
Download the video of your choice. I would personally use http://www.keepvid.com/ or google to find a different youtube video downloader. You want to save it as an MP4. I believe that one will let you do so.
Step 2:
You can easily remove the in-video sound in Windows Movie Maker by going to the Video tab and muting the sound there. You can then replace it with an MP3/WAV/Whatever of your choice. This is possible because the music tab has a seperate volume control.
Warning: If you use XP, you probably have an archaic version of Windows Movie Maker and the interface may vary greatly, rendering my directions as to where to find the video volume controls useless. You may however be able to find the archaic equivalent of this