Proof of domicile is an affidavit thing. Proof can include things such as your DL, voter registration, tax returns etc... Really not all that hard to prove.
Child of a K1 is a K2.
No reason to wait until the child is born to submit the petition. The newborn can be added later.
Processing times are not based on your home location as all I129F petitions start at the California Service center
Have you received local instructions? At most consulates the process for a k1 is applicant directed. I would not be waiting around expecting to ge given an interview date.
Who says its a discrepancy? With the significant amount of time between petition filing and the submission of the ds160 people's lives and circumstances do change