Yes you can use curl to do this. If it is a very large video then you will probably have to increase memory on your PHP installation or you can crash the server
if your php locks and can't close the connection. Being on shared hosting I would not attempt this. Your host techs will notice it almost immediately through bandwidth tranfers.
If you want to really do this, then you will need to be hosted in a cloud or a dedicated server and you will have direct access to your server.
And yes be careful about copyrights or you can end up with a hefty fine.
That's a good decision... writing the script is not a problem at all. I even wrote the script that will take out the ACC audio with the highest bitrate possible for MP3 conversion, but I can't get around with the TOS.
Although downloading the video first, will not make google block your domain in the first few months of operation, but as your site becomes more popular, the request will become noticeable to them to investigate the cause of excessive requests from your site.
There are few sites out there that are offering MP3 download, but most of them are hosted overseas and they can pretty much disappear at any time if any trouble with youtube arises. Unlike us, who are within the reach of the Big G, our options to play around -- or rather --> experiment with this type of programminng exploration is pretty much limited.