I haven't noticed any cool downs on the skills, and videos I've seen on perform make it look like you can hit notes in pretty rapid succession, so 8th and 16th notes should be entirely possible.

As far as the other, I highly suspect that the disclaimer not to perform third party songs is a way for Square to cover themselves and deflect liability onto the player if they do perform content by a third party. It will be interesting to see how that plays out, but I would imagine works that are considered public domain would be generally considered safe to play, as of course original pieces.

Edit: I would also say that public domain pieces aren't actually third party pieces since they don't belong to anyone.