English really needs another pronoun for cases like this!
Traditionally, if the gender was unknown, the masculine pronoun would be used. In a grammar book from 50 years ago, your examples would most certainly use "his." Today this would still be considered grammatically correct, but might be considered socially incorrect. These days, at least in the U.S., the "best" solution is probably to say "his or her", like this:
The one who had lost his or her identification card, couldn't come in. (This example bothers me a bit, probably because I think it's unlikely that the speaker would know that someone had lost a card and was not allowed to come in but would not know if it was a man or a woman.)
- Each person should take his or her equipment to the bus.
- We don't know the person, but we shall wait for him or her, wearing white shirts and holding daisies in our hands.
In some cases, rewriting the sentence into the plural would be a good choice:
"Passengers should take their equipment to the bus."
"Those who lose their identification cards will not be allowed in." Obviously, this is not always possible if the subject is definitely singular.
We would never use "it" for a person.
However, since our language has no pronoun for "a single person of unknown gender," people have begun to use "they, their, them" in situations like this. (Each person should take their equipment to the bus.) Technically it's not correct to use "they" for a single person, but its gender-neutral property is so useful that it is being used this way more and more. It's much easier than saying "his or her." I can't speak for everyone, but it doesn't offend me, and I would be very likely to say it myself in conversation. It would probably be best to avoid it in very formal writing or on English exams.
(In thinking about this, I'm not sure why the evolution of the language has favored using the plural pronoun as a gender-neutral singular rather than using the gender-neutral but inanimate "it" for people -- but that's definitely what has happened. It would be very common to hear any of your examples with the appropiate form of "their," but never, never with "its.")