My Business is to have an to sell products online (online store). My business name has just been approved and I have also purchased the company desired domain. I am in the process of taking my first steps towards implementing the structure of my business and I have the following questions on my mind:
People in the USA won't be able to pay in Rupees, they will insist on using US Dollars. So, you can get a US Dollar denominated bank account, or you will have to pay for conversion, and be exposed to currency fluctuations (you can hedge that, but it is non-trivial). A global bank like HSBC, for example, can help with the bank account.
As for your second question, it is impossible to answer without knowing what business you intend to embark on and your own capabilities, etc.
There's really no short answer to this. Each online store is different. It does sound like you need to read an article on the overall process of running an online store. Do read http://blog.wixelhq.com/getting-started-with-e-commerce-a-guide-to-starting-an-online-store/ or download a PDF here: http://www.practicalecommerce.com/articles/1201--Getting-Started-in-eCommerce-PDF-Now-Available These types of articles cover the basics of running an online store - in so doing you will have knowledge of the payment landscape as well as the development and marketing considerations.
I would advise you to hire a developer to set up your ecommerce site online. That way you don't need to worry about the technical aspects and concentrate on your business work.
For payment gateway there are a number of payment gateway service providers in India. I have given below for your reference. I can't yet post links, but you can google them.
Using a service like Shopify would probably help. Since the payments would pass through them, they'd deal with those issues and then you'd get an aggregate payment. The key value point of most such services is that you can manage the store without significant technical know-how (although you might want to contract a developer for customization or setup).
A more extreme case would be to use a fulfillment service, e.g. Fulfillment by Amazon. You would then send an aggregate shipment to them, they'd ship to your customers, and then send you an aggregate payment. You might need to jump through some hoops to use any particular fulfillment agency, e.g. set up a US corp. and coordinate with an importer.