Understanding IP fragmentation

I understand IP fragmentation is bad. I want to understand better how to prevent it.

1. If all nodes on a given network share the same MTU, does IP fragmentation ever happen?
2. Let's say PMTUD is broken. If I add encapsulation (say IPsec or GRE), does the router performing encapsulation also perform fragmentation for any frames received that will total more than its outbound interface's MTU with headers (and that don't have the DF bit set)?
3. Let's say I set the router's outbound interface's MTU to 1400. PMTUD is still broken. Is the router still performing fragmentation? I assume so, because I don't know how any devices behind the router would know about the reduced MTU. So does reducing the MTU on the router even help prevent fragmentation? Surely I've missed something?