Yes, but I would expect that to be part of onboarding: Hey user, this is your pirvate key, store it safe, and here is you recovery key, store that somewhere else. Of course clients should offer that as a function to use at any time. Also the keys could be generated from easier to record seeds with enough entropy.
Discussion
Yup and yup.
Although I would prefer clients not to bother you with something like that until you’ve used your account “enough” but that’s an implementation detail and up to clients to create different onboarding flows