Any suggestions of what to do during a 5 hour layover in Frankfurt? I've read that it's possible to go into the center city in that amount of time, but I think I'd feel rushed and afraid of not getting back in time. Notes about check in at Frankfurt say to allow at least two hours. Since we'll have boarding passes, is the time required shorter or does security take a long time there?

We're booked through United, but flying Lufthause. Can we use day passes for of the lounges there?

Your intuition about going into Frankfurt and back is correct. Find a lounge. Lufthansa lounge access depends on class of ticket. If you're in business, you're in. If you have a United one-time pass through your CC I don't believe it will be accepted. If not, I would purchase a pass.

No experience of lounges but you are right to stay put in the airport. Immigration can take a long time and Frankfurt is an absolutely massive airport. It's not inefficient but you may find yourself running to the gate you want just because of how things are laid out. I fear that if you left the airport, allowing for the time you need to queue and get to the gate, you would not have enough time to see anything in Frankfurt proper.

There is a bit to see and do in Frankfurt, plus it's a great gateway for the middle Rhine gorge. Give it the time it deserves and do a Germany trip

Lavandula

I'm glad you're not going into the city. A lounge pass would be good for such a long layover. You can have a shower, a meal and some quiet time.

We transit through Frankfurt several times per year. With a five-hour connection, I wouldn't leave the airport. You'll have to go through passport control on arrival from the US, so allow time for that.

From this page on the LH site, you can search for info about lounge passes. Depending on time of day, the LH lounges can get quite crowded. I would not buy a pass in advance, as you won't know until arrival how much time you actually have or which area/gate your connecting flight will be departing from.
https://lufthansa.travelintegration.com/lounges/search?