While this isn't a direct answer to your question, you might be interested in this thread that explains how identity was verified in regards to bank transactions before the advent of photo IDs.

Age and citizenship have a little bit different answers. Depending on the state, birth certificates have been in use in the U.S. since the last couple decades of the 1800s. Before that, there were censuses you could refer to listing all residents of a household going back to 1850 (and ages but not names all the way back to 1790). These were obviously not fool-proof. Someone with more knowledge on the Civil War, for instance, can probably provide you an informed answer on how prevalent it was for underage soldiers to pass themselves off as fighting age men.

It's helpful to know, though, that age and nationality requirements in the U.S. pre-20th Century were much different than they became later on. There weren't bouncers checking IDs at the door of the local bar. Child labor and immigrant workforces were common in the early decades of the Industrial Revolution. Employers weren't exactly concerned about doing background checks.

And before that, before the building of the railroads, the issue wouldn't often come up since most people you would encounter, would be people who lived nearby so age and nationality would be difficult to lie about. Even then, there were few situations where you would ever need to prove your age or nationality. Military drafts would be the big one.

If you have a specific question under what context someone would be asking for proof of age or citizenship, you might get a better answer.