Can I start a LLC on an H1B visa and show profit as not self employment income?

I assume you're talking about starting an LLC in the United States.

If law says you can't work for a company that isn't your sponsor, doesn't that only pertain to work that is here in the US? Surely it doesn't cover the whole world. And if so, why not just create the corporate shell in your native country?

Many assumptions made above, but I'm not a lawyer, I don't know the first thing about H1B visas, and I have no idea what starting a company in your country is like. :) I'm also assuming the company your H1B is through is also fine with you doing other work.

H1B prohibits working for any company that isn't the company that sponsors H1B.

You can create an LLC and hire other people, but you cannot work for it yourself.

So in he sense you care about: no.

See e.g. https://news.ycombinator.com/item?id=8198916 or https://news.ycombinator.com/item?id=4534287

Should be fine. If you own the business, it is the capital gain, the same as stock dividends.