For a "sole proprietorship", is it necessary to register a company at all for doing what the OP asked (starting a website as a side project), or can he just do it as an individual and just pay taxes on the profits as regular income?
PS> I am thinking of doing something similar to the OP, and not sure if this complicates things, I am on a H1 visa.
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)
* Your city-of-residence probably requires you to file (and pay annually) for a business license, even if you are doing a purely online business.
* If you accept payments in any name besides your own (e.g., a website name), and you don't have an LLC/corporation/partnership, you probably need to apply (and pay) for a DBA (no, not that kind of DBA - a "Doing Business As"). Accepting payments in a name other than your own without this could result in prosecution for fraud.
In my personal IANAL opinion, the LLC (Limited Liability Company) is the best "side projects for profit" option. Relatively simple and inexpensive formation (order of $100 w/o lawyer), limited liability (how limited? courts will tell), no "double taxation" from a corporate structure.
PS> I am thinking of doing something similar to the OP, and not sure if this complicates things, I am on a H1 visa.