So do you use the construction "eu estou com" when you're talking about having something in your possession at this very moment?
For example, if you're saying "I have my passport (with me right now)," you would say "eu estou com o meu passaporte."
But if you're saying "I have a passport (in general)" you would say "eu tenho um passaporte."
Is that correct, or am I way off base here?
Yes, that's right. You could also say "Eu tenho passaporte" (in general)
Eu tenho um livro = The book belongs to me.
Eu estou com um livro que leio agora, mas o livro não é meu = I have a book that I'm reading now, but the book doesn't belong to me.
Estar com = temporary
Ter = permanent
Franz, it's just another way of saying the same thing, although there is a slightly different emphasis. In (American) English we might say something like 'do you have your passport on you?' / 'It's not on me.' to mean you have it right now. Do you have any cash on you? It's just one of those things...
The 1st hover hint for "eu estou com" is "I have had," which would sound natural if the sentence continued on (such as "I have had my passport for several years.") But, given that Duolingo is so fond of sentence fragments, I used "I have had" in my answer, only to be marked wrong. "I have my passport" is much more natural. Then, why, Duolingo, are you trying to confuse us in the hover hints? WHY?!