I agree with @Baiju that you need to have a discussion for this, preferably with a partner near your location, instead of just asking question.  Because even simple words like invoice or sales can have different meaning for different people (I had had a client that regards only invoiced sales as sales, so sale order has no meaning).  So, below are my best attempt to respond to the requirement list, but do take it with a grain of salt.

1. Invoice generation - We should be able to pull Odoo invoice on our website where user will pay it. After payment is done, Invoice should be updated to Paid. Odoo basic accounting functionality covers invoice generation by default.  Sale functionality provides the feature to generate invoice from Sale Order.  After payment is fully made OR refund is performed on the invoice, invoice will be updated to Paid state.  Odoo has some functionality to put invoice up for your customer to view (Partner portal).  However when it comes to payment (I presume you mean online payment) there is, AFAIK, no official module for that.  There are some community modules that provide integration with some generic online payment provider like PayPal.  However if you opt for a more local provider (even VISA or MasterCard), my experience tells me that you might need to build something. 
2. Product with multiple variants. Odoo by default enables you to manage product with multiple variants.  There are also some community modules that provide extension to that (e.g. product_variant_multi).
3. Ability to setup recurring billing for a product for phone orders and take payment on odoo itself using firstdata global gateway or authorize.net.   There are some community modules that enables you to setup recurring invoices. On Phone Orders, not sure how you are planning to integrate this (IVR? Call Center?), but I'm not aware of any specific solution for phone orders or firstdata global gateway or authorize.net.  Also see my answer for 1.
4. Ability to void/refund a payment on odoo itself and send customer a receipt for it.  Odoo does provide a capability to reverse a payment, though if you really want a hands-off operation for the refund (one click that does all), it need to be developed.  Also, receipt sending need to be developed.
5. Logging customer calls.  You can either do this manually using the CRM functionality or integrate using Asteriks.
6. Add comments to customer details. This is a broad subject.  But you can put information in Internal Notes of the Customer, or note it in the conversation.  This is on top of the various fields that are available for a Customer (partner).
7. Providing customers an online helpdesk system.  Web building is a new feature in v8.0 which you can use to build almost anything, like this forum site.  I'm afraid I'm not too well versed in that feature.
8. Multi store setup. We should be able to see which website did the sale. Both websites have different products. I don't have experience in this.  So, I wouldn't provide any answer.
9. Customer info should show each order details, IP address, date time etc of order, call log etc.   There has been a several discussion on obtaining client IP address in this forum.  Short anwer: you need to develop it. I hope you are aware that customer's IP address may change.  Date time are logged when a record is created.  From Customer view, you can pull (by a click of a button) the Sales and Calls related to that Customer. 
10. Order management - we should be able to see total sales/orders for the day per site.  I don't have experience in this, however in general in Sale Order list view (or any list view in that matter), you can filter by almost all fields available.

Hope my answer help.