Depends heavily on the time of writing whether all those things took place. The closer to get to modern time the more likely it happens and even some of the live action adaptions have put different spins on how and why it took place, like with the Kents asking Lionel Luther for a favour expediting the adoption process in Smallville in return for saving Lex during the meteor shower that brought Clark to Earth in the first place in the show.

As mentioned in my other comment some version just didn't bother because they would just claim someone else left the baby on their door in the middle of the night which happened in the earlier parts of the 1900s and far fewer people gave a shit about looking into how legit that might have been, nor were busybody types to then hound someone over the paperwork. Ironically that kind of busybodying is one of the key traits often written into Lois Lane and why she ends up figuring out who Clark is in many stories because she's one of the few people who bothers to do what would have been highly unlikely in the original stories. Nowadays it's mandated bureaucratic handling that in hindsight makes stories like Superman seem less believable when the entire point of older stories is that it was literally a different time and people behaved very differently, not a perpetual progressive state always thinking about the $CurrentThing, which often ends up turning fantasy stories into shit as it did when Superman comics tried to do climate change plots.