There may be local rules which require some form of business registration, but in the US the typical case is that the first -- and only -- time the government hears of the existence of a sole proprietorship is on Schedule C attached to the individual's tax return, which documents its income and expenses, calculates the profits, and tells you what other pieces of paper to copy the profit number onto to figure out how much tax you owe.
Sole propretorship == doing it as an individual.
This is not legal advice: if you're physically in America when doing the work, the IRS probably doesn't care about your citizenship, you owe it money on your sideline income. It may not be, strictly speaking, legal to work for someone other than your employer (yes, including yourself) while you are on a H1. In practice I'd bet you that this is commonly overlooked.
About the H1, you are right, friend of mine lived in San Fransisco on for a couple of year, on an H1 visa, he had a full-time job, and had a small revenue from his website, paid his taxes accordingly, never had a problem.
(It's actually the same in Canada, i'm in locked visa, and i do contracting from time to time, never had a problem)
Sole propretorship == doing it as an individual.
This is not legal advice: if you're physically in America when doing the work, the IRS probably doesn't care about your citizenship, you owe it money on your sideline income. It may not be, strictly speaking, legal to work for someone other than your employer (yes, including yourself) while you are on a H1. In practice I'd bet you that this is commonly overlooked.