>>Question 1. Does St. Anthony's still exist <<
google says yes, it still exists.
>>Question 2. Is it possible to get a copy of Henry's membership records to these societies?<<
That depends on what records this parish archived. Since these were not sacramental records, they might not have been a priority for the parish to keep. Only they can tell you.
>>I'm wondering if his place of birth might be listed as we have no idea where his home parish was in Germany.<<
Was he married in a Catholic Church in Wisconsin? Depending on the priest, marriage records could have good detail, including naming the home parish of the participants.
Was he naturalized? If so, his declaration of intent might have the name of the town/city he was from. Did he have other family that immigrated? Researching associated family members can help you find clues to your direct line.