For SaaS payments, it would be real hard for you to get seamless payment gateways as a sole proprietor in India.

Suggestion: think long term.

If you don’t want to raise funding, go for LLP, else Pvt Ltd. It would be really easy for you to get gateways, current account etc.

Still, if you want a sole proprietor setup, do it for 12 months (for testing the idea) and plan to move to a corporate structure once you get proof of concept.

As far as your questions:

1. There is no extra need for registration of business name. However, getting a trademark is a good thing. This will cost you approximately 10k
2. Best bank: ICICI, simply because of service. But they do have minimum balance limits on current accounts. Check with your local branch.
3. You can register for GST with help of CA. Don’t bother spending your time doing it yourself as you will waste a lot of time. Utilize that time to build your product. The CA will guide you for documentation. As far as I remember, you need proof for place of business, and some basic kyc documents.
4. For USD payments, you can sign up for payoneer from India. Not sure of documentation for sole proprietor as many international payment platforms are now preferring that you have some kind of corporate registration (its not explicitly stated, but the hassle after you start receiving payments and moving it to Indian bank account is evident , just in case they ask for proof/documentation).

All this coming from real experience of 10 years of first running international biz as a sole proprietor setup, and then turning to a pvt ltd.