Actually, your immigration status doesn't matter when it comes to investing.
If you don't have a work authorization, you can't actively participate in the management or operations of any company, paid or unpaid. That's where the confusion often arises.
But as long as your investment is passive, it doesn't matter where you live or who you are. You get stock, you write a check. Anyone is allowed to do that.
I don't see why not.
I'm not a lawyer, but I've been the secretary of three corporations, and in 1985 read the entire Texas legal code on corporations. As far as I know, there are no legal limitations on the nationality or immigration status of investors.
You should have the startup's paperwork (partnership agreement, incorporation papers, etc.) drawn up by a competent attorney, and they can confirm whether my opinion is correct.