Would renting out a house be considered as work or are there any other issues?
All taxes will be paid, of course.
Yes, absolutely. The same way as an H-4 would be allowed to buy stock and otherwise invest in the stock, bond and commodities market (you would lose your sari.)
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I would say you can. I would not see it as work.
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Yes. That person can do that as long as they do not create a management team and start working there.