Visa's are by definition documents providing authorization for (re)entry and specific kinds of work. If you're not inside the country's geographical boundaries, you are not in their jurisdiction and hence don't need a visa.
Now, taxes though... that's a different beast.
Easy: you invoice from wherever you're based. Exempt from tax in the US, taxed in your home jurisdiction. Expenses tax deductible.
No, there are multiple options. If you are a contractor is one way, or an own Delaware LLC that is used for invoicing, you just need to hire international tax accountant to be good with both US and your-country-of-tax-residence's tax offices.
Of course not. I'm not sure why would you even think that.
Why do you seem so certain of that?
I live in Germany, work for an SV company, and a work visa in Germany is absolutely required.