How do you know whether to use "svou" or "moji" and their variants?
If the owner is the subject of the sentence use the reflexive forms.
It accepted "Já mám moji manželku", but I don't believe it should have, unless the reflexive forms are optional.
The reflexive form is normally much better. The non-reflexive forms have their place when you stress you have YOUR wife and not someone else's.
I've seen "svou" and "svoji" used interchangeably when the possessive precedes a feminine singular noun.
(Hledám svoji ženu in a preceding sentence, and Mám svou ženu here.)
Is there a difference? If so, what is it?
both are standard. the hard adjective declension (svou) is more formal.