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1) could my iPhone sync with my iPad only and her iPhone sync to her iPad?

Yes. You'll just want to make sure you and your wife are syncing to separate places: 2 different Omni Sync Server accounts, for example.

2) how important [is it] to have a persistent internet connection?

OmniFocus does require internet access to sync, and to display web pages (if you store links in the notes of actions or something). So, until you have multiple devices syncing together, it should work just fine.

3) Bonus, unsolicited advice:

Even if you each only have the app on one device, you may want to go ahead and set up syncing, so that there's a backup copy of your data in case something happens to your iPad. (For that purpose, syncing only when you're at home would work just fine.)

Does that help?