Baggage rules

1) You go by the operator of the aircraft...United rules on United, Lufthansa's rules on Lufthansa.

2) No. But you will go through exit immigration in Frankfurt.

1. No. Zagreb to Frankfurt is a domestic (in Schengen) flight; as long as 1 ticket they should be checked through, you can confirm when you drop them off.
2. Usually it's the owner of metal you're flying on.

While typically baggage rules of the plane you are flying on apply, on your way over, since you start on United, when you change planes it would be unusual for them to be too strict when you move to the Lufthansa flight. However, on the way back, expect that they will apply Lufthansa's rules

Who ever is doing the check-in controls your bags. And Lufthansa can be very strict for weight.

If your carryon meets United restrictions that does not mean it is ok on Lufthansa. You best check their restrictions.

1. Some airlines have reciprocal carryon agreements. But I don't know about Lufthansa and United.
2. Often the limitations will be printed on your booking confirmatio email if you purchased from the airline. If it shows the United limits then they have an agreement.
3. No matter what the rules, the agent at the gate is the final word. Don't argue, there are others waiting to get on the plane.
4. None of airlines are as observant as they should be. But if one person fails a check it ends up on 100 websites. If 10.000 get through no one posts.