Hmmm. So you would use "maji radI vs. maji radY" because of the one male noun? Is that the "default" when the sexes are mixed? Conjugate in the masculine?
masculine animate has the priority
Why is this wrong? "František a Kateřina jejich jména mají rádi"
First I do not like jejich. That really sounds like it is about names of someone else. You shoud use the reflexive pronoun here. However, that IS accepted here.
The word order was not accepted here. So I added it.
It is fair to say the svůj (in all of its forms) should be used whenever it's applicable?
Yep, it is better.
And why is this also wrong?
František a Kateřina sveho jména rádi mají
It must be svá as it is accusative plural, svého is singular genitive or masculine singular accusative.
It looks like I wrote "sveho" when I intended the write "svoje". Would svoje have been accepted?
Svoje is fine and less formal.