Halfwayback,
There have been a few threads about the technical aspects of copying such a file to your hard disk - searching on You Tube in this forum should locate them.
As regards copyright - this is a minefield! You are possibly talking different jurisdictions, with probably different laws. The nature and type of content may also have a bearing on whether copyright applies.
I am certainly not qualified to advise you, but I believe that the uploader of the content would probably be the holder of the copyright, so a message to them might elicit the permission you require (I think that each uploader can be contacted via youtube). But it could also be that the conditions of using youtube allow youtube to jointly or solely own the copyright. And there are certainly loads of clips (e.g. of tv shows) that the poster is almost certainly NOT the copyright owner!
I don't think that it can be described as "public domain" simply because it is freely accessible over the internet, btw.
SD