I've tried various ways of downloading Youtube videos. Most seem to work for a while and then, as Youtube changes, cease to do so. I invariably seem to come back to Ant Video Downloader which works with nearly all video-hosting sites (not just Youtube):
http://www.ant.com/video-downloader#video_downloader
Once you've got a video stored on your hard drive, you'll want to watch it and the most frequently-recommended program for doing so (which is about a zillion times better than Windows Media Player) is VLC:
http://www.videolan.org/vlc/download-windows.en_GB.html
(Use 'Installer Package' if you've got a 32-bit version of Windows or, obviously, '64 bits installer' for a 64-bit version).
The reason I've mentioned VLC here though is that it's got some useful video-editing facilities hidden in its 'advanced' controls, which will enable you to clip extracts from your files. (That's how I do it. It's dead easy!):
http://www.guidingtech.com/8802/use-vlc-as-video-cuting-tool
Any decent DVD burning software should then enable you to put your videos onto DVDs. My own recommendation is Ashampoo Burning Studio:
https://www.ashampoo.com/en/usd/pin/7110/burning-software/burning-studio-free