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Will a 'shrine' make family members fertile or does it have to be a 'temple' or greater?
Yes,
Some factions have the fertile shrine.
https://forums.totalwar.org/vb/showthread.php?t=37949

Temple will do the trick just fine.
I Imagine you get more the change off fertility the greater the temple is.


Also, is fertility influenced by movement points? Does the governor have to have 100% movement points to get...fertile?

OK.
That’s got nothing to do with it!
Just keep you’re governors in a city.
Don't move them during turns.