Yes, "own", because a domain is just rented out, but better than being owned by Google.
Yes! That was one of my main concerns. I have my own email domain, but it is a .xyz domain and I have had some people struggling with the QR code on the back, some QR code apps don't recognise webpages in the .xyz domain (looking at you Xiaomi camera app). They just recognise it as plain text. So you have to copy and paste it into your browser.
So I am a bit self conscious about having to explain that that thing is an email address. Will probably add it on the next batch though.