There is no problem with any of it from the immigration stand point. I posted your quesiton into business section for more specific discussion.
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I agree with my colleague.
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These are the corporate and buisness law issues, please meet with a relevant practitiioner to evaluate your plans.
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If my immigration colleagues have no objections as to immigration law, then you should be able to set up the LLC in Illinois or Delaware (or anywhere else, really). If you leave the US, there's no reason a non-resident can't continue to own the LLC. Your tax issues become a lot more complicated, though.
I'd recommend putting together an advisory team that includes an immigration lawyer, a corporate/business lawyer, and a tax specialist (lawyer or accountant).