1. If you are self-employed, you pay state income taxes based on where your permanent residence (domicile) is located, or where you physically perform the work, if that is in a different state. While New York can sometimes tax non-residents who work for NY-based employers, that does not apply for self-employment.
2. If you are a single-member LLC not treated as a corporation, then you are taxed as a sol proprietor (disregarded entity). As such, #1 applies and your income is not NY sourced unless you live or perform the work in NY. However, there may be annual legal fees that I can't speak to. You may want to talk to a NY-based accountant.
3. You should talk to an attorney about that. LLCs are creations of state law, and each individual state may have different laws, rules, benefits, and costs of being an LLC.
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