Yes, there is a free Youtube app available from the PSVita's PSN store.
Youtube can be accessed by downloading the free Youtube app from the PSVita's PSN store.
The Youtube application has been withdrawn, one of Youtube's major updates caused a lot of compatibility issues across a range of devices, breaking them completely in some cases. In the case of the PSVita's, they decided not to fix it on the app itself. Instead, on PSVita you should now access Youtube directly through the browser.
The PSVita does not use UMDs, your games either come on the new PSVita Game Cards, or are downloaded from the PSN.
The PSVita is not region locked, a US PSVita can play an EU PSVita game as would be bought in the UK.
Yes, the PSVita has physical games, small cartridges called PSVita Game Cards. The PSVita has a slot at the top for them.
There are some free downloadable games, yes, but the PSN is mostly a store to purchase digital versions of games.
No, at this moment in time, is it not available for PSVita.
Yes, the PSVita's screen is HD.
The model is PSP Slim and Lite or PSP 3000. They are the same thing.
Yes subscribing on Youtube to anyone's Youtube channel is totally free. You can simply subscribe it for free through your email address.
It is within the 'apps' section on the PSN store. You could also search for it using the search function.