Or not being able to play a board game, because it says “ages 9 - 99” on the box.

I’m not sure about this. Ever heard the phrase “the past is a foreign country”? Living through time would be like immersing yourself into a new country every couple of decades. You could even lessen the blow (and would probably have to in order to remain anonymous) by frequently moving around the world. People tend to give newcomers a certain amount of slack and with the enormous amount of knowledge and experience you would gain over time, you can easily and quickly immerse yourself in any new environment and adapt to whatever is “it” now.

I would only accept this kind of deal with the Devil if I had the power to turn people I care about into fellow immortals, which comes with its own pitfalls I’m well aware of (imagine holding a grudge for thousands of years), but still.

Probably, but it depends on the person. I stopped caring about some relationships that ended after a year, but I’m still thinking about others decades later.

Then again, you have billions of years to come up with a solution to this problem.

Not everything is on the Internet. Even most information from as recent as the 1980s isn’t. A not insignificant percentage has been digitized (this is an ongoing effort), but this doesn’t mean you can just google it or access it through a website.

If you know where and when your ancestors lived, hit up local, regional and national archives. Church and municipal records, national surveys, newspapers (useful for announcements of births and deaths alone), school and university records, etc. You’d be surprised by how much you can find this way. If you’re living too far away to visit in person, give them a call. Archivists are very helpful people by nature and occupation.