I use Videoder myself; that app is no longer on Google Play, but you can Google their website and download it for free.
I would assume not. Remember that Google owns YouTube now, as well as Android, so the odds are damn slim that they're going to let anything onto Google Play that will let you download YouTube content.
I have nothing, but remember that Google a) hardly controls the Google Play store, and b) easily allows installations from other, completely uncontrolled sources.
I did say "used to." Now, the app is only on the developer's website which is where I got it after I installed it to my new phone.