Ultimately the law in which you are looking for will differ depending on the origin of the country in which you register the company.
Hopefully the below will shed some light on your questions from a UK perspective.
1. Ultimately you will need to register the business as your profits will need to be taxed. If you register a limited company you will have an annual return to process. Alternatively, you could setup as a sole trader, where your accounts would not be public, but you would have unlimited liability.
From your membership site, what will you be selling? You may find that your products/services dictates the amount of risk you may have and decide the company structure for you.
3. If this is a side project and you are only forecasting a few thousand pounds, using something like paypal is fine. If you have an excellent product/service and you are looking to grow a successful/large business, you should speak with your retail bank to get setup with a merchant account and look at gateways such as realex, ogone and sagepay.
4. Because you are selling products online, your customers will be using credit/debit cards to make payments. Visa/MasterCard have a chargeback window of up to 6months for customers that have been miss-sold products etc. You need to have water tight terms and conditions on your website and ensure your goods are fit for purpose to protect your businesses interests. It's difficult to advice against refunds when I don't know what the business is going to be doing.
For the best responses to your questions, you should detail what you are planning to do
Hope this helps!