I did file last year's state income tax form, as I use a family member's address for all mailings. But as far as any presence goes, I'm not in the country, and technically not a state resident. I also dont have a business incorporated in the US and am doing it under my own name, for now.
Do I need to charge state sales tax? Does that change if my company is registered in, for example, Singapore?
Sales tax is not federal tax/IRS thing. The IRS only deals with "income" tax. Sales tax is dealt with at the state level or lower. Almost all states impose a sales tax of some kind or another. Even some cities within a state may impose their own sales tax on top of a state sales tax too. Some states require you to pay them a sales tax if you sell any product in their state, regardless of where the business may be located or where the product comes from. So that has to be figured out for each state where you ship product to. Unfortunately, we can't help you with that here.