"František and Kateřina like their names."

[Daniel-U]

Why is this wrong? "František a Kateřina jejich jména mají rádi"

[VladaFu]

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.

[Daniel-U]

It is fair to say the svůj (in all of its forms) should be used whenever it's applicable?

[VladaFu]

Yep, it is better.

[Daniel-U]

And why is this also wrong?

František a Kateřina sveho jména rádi mají

[VladaFu]

It must be svá as it is accusative plural, svého is singular genitive or masculine singular accusative.

[Daniel-U]

It looks like I wrote "sveho" when I intended the write "svoje". Would svoje have been accepted?

[VladaFu]

Svoje is fine and less formal.