My experience is that United will prevent you from checking in through their website (giving an error that points you to check in with the carrier operating the first segment). The Lufthansa website may or may not check you in for the United flight, depending on how your record is set up. This doesn't matter, though, as you'll be fine in FRA (I've made it through immigration without a connecting boarding pass, telling the officer that I need to go to the United gate to get it). I've also found that check in for the United flight becomes possible through the app/website as soon as the partner flight closes. That is, if you have internet access on a mobile device during pushback (and are permitted to use this mobile device in a transmitting mode), you can check in for the United flight once the LH flight has departed.

I'm not familiar with the passport error that you mentioned.

Enjoy the trip!