I buy a lot of old photos off of ebay. Most of these photos are photos of ships, and date back to the early 1900s or sooner. These are not photos taken by a professional photographer. These are one-of-a-kind photos that were taken by someone unknown, and the person who sold them to me is not a family member of the photographer. As the person who owns the actual photos, what are my rights? Do I own the copyright, and if I post them online, do people need permission from me to use them? Thanks.
The only way you could own the copyright (as a non-creator) is if you have purchased or inherited that right. You might have inherited the right to a photo taken by Aunt Tillie, but there would have to be an explicit transfer of ownership of the right from the former owner to you.
Failing that, the question is what the copyright status of the photo is. The first question is whose law you are dealing with: US? UK? France? Under US law, there is a higher probability that copyright has expired. Assume these are US photos. Then the next question is whether the photo was published: let's assume it was not (it if had been published that could work to your advantage because but you would have to prove that it was published). Then you must determine whether it was registered: again, assume it was not. The point is that copyright could have expired under certain conditions, under US law. Unregistered unpublished works are protected according to the provisions of §302 of US copyright law (as specified in §303), where the rule is "70 years after the author's death". Then you have to know when the author (photographer) died. If it was before 1949, the work is not protected by copyright.
In short, you have to know a lot of facts about a work, in order to be able to know whether it is still protected. This pamphlet by the US Copyright office explains the rules pertaining to copyright duration. This is relevant to the prior question of whether it is legal for you to post these pictures online.
Assuming this is the USA, the copyright for those photographs would have lapsed at a maximum of the date of death of the original authors of the work + 70 years.
Of course, it is unlikely that anyone would ever come out and sue you arguing that they are the original author, or is the estate of the author. So I'd just go ahead and reproduce the works if you'd like
No, you don't own a copyright if you didn't take the pictures. You have to apply for a copyright, like a patent. It does not just come about because you own a picture. https://www.copyright.gov/help/faq/faq-register.html