depends on your definition of "legitimately". there's not much explicit pet play, but some randos might write "pet" on you, and you can get a paw tattoo, and a hunter(ess) in the woods can nap you and put you on a leash, and you can get kemono transformations, and moo at robin, or request headpats, but there's no good "owners" npcs dedicated specifically to the kink. like, there's enough stuff to facilitate role-playing, but you'll have to use imagination.
no. no need to be at 7/7. every midnight - you gain 1 purity point, then 2 more if you're a true virgin (temple doesn't count), then multiply all that by the number of holy clothes you're wearing + 1. if you're at 1000 purity after all that - gain a point, otherwise - lose it. that's enough buffer that you'd have to masturbate to completion at least 3 times daily to offset angel purity gains. fail at that - lose 2 days of trying to lose angelicness the slow way. i guess it's worth mentioning that you don't turn into fallen angel that way, and can still gain the tf back if you want, as long as your virginity (this time including the temple one) is intact
angel requires commitment to virginity and full purity to keep it. demon requires even stronger commitment to daily impurity since even 1 point at midnight (before daily gains, obviously) will result in 100% stress and consequent passout. fallen angel is the middle ground option for when you're bored of angel tf, but don't wanna commit to demon either, while keeping some of the divine tf benefits. there's no need to care about virginity, nor daily purity, because even though you lose 9 points daily - attending saturday mass restores 70 points vs 63 that you lose in a week between. and that's before modifiers, which can cover a lot of sinning. admittedly, the perks are not as strong, but you do still keep the wings and some perks to combine with non-bird tfs. staying above 900 purity for 30 days straight will even give prettier wings and a "disparage" ability, which is basically a "mock" action with 100% chance to target the right insecurity. and once achieved - you don't even have to maintain 900+ purity after, the requirement then becomes just 10+ purity at the end of the day, as you're a forgiven fallen now.