I own a fairly early Amiga 500 with Kickstart 1.2 ROMs, Commodore Logo key and the OCS Agnus 8371. Is this system completely compatible with Workbench 1.3? Or does WB 1.3 require Kickstart 1.3 ROMs? What kind of problems, if any, can I expect when running WB 1.3 on it?
No, you can and, more importantly, it is fully supported to run Workbench 1.3 on a Kickstart 1.2 machine.
This was rather common back at that time, because Workbench 1.3 brought some nice new features (like the RAD: drive, better printer drivers), and at least in Europe the first batches of Amiga 500 and Amiga 2000 arrived still with Kickstart 1.2.
On almost the same subject you can see question What exactly are the differences between Kickstart 1.2 and 1.3?