It’s very cute! It could be domestic, especially if it’s all alone. Here’s a good article on catching pigeons if you want to try: https://www.pigeonrescue.org/faqs-2/how-to-catch-a-pigeon-or-dove-in-need-of-rescue/.
Yes. Though, if this is a domestic pigeon, it is not going to be safe outside without a flock, so would have to be contained in an appropriately sized enclosure if you intend to keep it permanently.
He may have been an imprinted pigeon, meaning someone probably found him as a baby, hand-raised him around people, and then released him. Unfortunately, imprinted pigeons don’t associate with their own kind, so are often in danger in the wild.
Chicken scratch grains would work temporarily, but they’re not a good long-term diet. I would just get a pigeon seed mix instead of trying to create your own, I order mine online.