Odin stuck at "File analysis"
I did 3 things, tried again, and it worked (I mean it got past File analysis). I am not sure which one of the 3 things did the trick, unfortunately. You might want to try them all.

Run SAMSUNG_USB_Driver_for_Mobile_Phones.exe
Use a different USB port, for me COM7 instead of COM6.
Get the ROM for my phone's region. It is binary equal to the file I tried yesterday, though, so I doubt it changed much.