"Just curious, can someone with no visa (or wrong visa - tourist, F1 international student, or even with H1 Foreign Worker) create a Corp or LLC in US?" -- Yes & No. Any foreign national can create a Corp/LLC in US. That goes through foreign national corporation rules. There are laws that govern and what industries you can operate and what businesses you can run. If you are on a non immigrant class visa (B1, B2, F1 etc), you are treated as a foreign national and thus will be governed by those. For e.g. you cannot take money out of business within US etc. You can create and own a Corp/LLC if your class of visa is "immigrant" (e.g. H1B, L1 etc.). However you cannot work for your firm. For example, you can be legally in the country on H-1B working for an employer and then provide consulting/guidance to people in the neighborhood on how to garden and charge them. You can route the charges through a business and have protections that LLC/Corp offer. You can code in parttime and route the revenues you earn through the firm (however your primary employer might come back and claim any of your inventions as theirs.. be careful). You can hire US Citizens or permanent residents and run the firm as an independent board of director and take money out of business. All of this is possible for immigrant class of visas. The only catch is that you cannot work for it. Remember that if you hire an american citizen as CEO and then file your H-1B through your own firm, you can surely forget to get a permanent residency. You cannot file your own employment based green cards no matter what revenue you make or killer app you developed. The law is very clear there. There is severe self interest and you have to prove that there are no US citizens to hire for that post and when you own the firm, you have self interest to reject any legitimate resumes that come your way.

"If so, how can the founder of the LLC or Corp continue to stay in the US? Does it mean creating an LLC or Corp in US guarantee the founder a work visa? Or they still need to find visa sponsors?"

The founder or the LLC or Corp have to stay in US using the same process they used when they entered US. They can switch firms (not to their own) using the same technical skillsets or any additional skill set they gained through training on their previous job. There are very few ways for attaining green cards and then citizenship. Keeping the lottery and the freebies that military doles out, it largely boils down to specifics like marriage, employment and investment. There are clear cut rules for each category and Startup founders do not fall into any of the slated categories (because these laws were developed pre internet startup boom). There have been bills proposed and killed in the congress. Nothing has changed much since the last 2 decades. Keeping all non work related visas out (which includes visitors, business visitors, marriages, military programs, lottery program etc etc), the only viable option for gaining permanent residency and in future citizenship is through employment based categories (EB1, EB2, EB3, EB4 & EB5). None of these categories have anything for founders. The intended founder of a firm have to work for other employers until they get their green card and then start their own firms. Or work as co-founders in a private firm (without disclosing their co-founder status and stock ownership) and file for employment based visa and then kick start the green card process. Finally work for an employer and work part-time without telling anybody, make a killing (i.e. >$1,000,000.00) and then invest it back using EB5 (investor) category. Investors get 2 year temporary green card and at the end have to show fulfillment of the needs laid out upon which temporary will be made permanent.

-- I've been on H-1B visa for over 10 years and still waiting for my green card. I've started about 4 firms like restaurant franchisees, real estate, importing goods, tech body shopping & websites. I've closed down businesses when I wasn't able to devote much time to those. Starting a business and running a company is an amazing feat. But don't be fooled into an easier route of permanent residency through this. Law is very clear and you might loose your opportunity to work or stay in US if you play games around this. I'm working part time with 2 firms incorporated by US green card holders while I continue to work for my firm 8 hrs each day.