What is involved with WordPress and trying to develop our commerce site that way? Will it integrate with our CounterPoint register and inventory system? Does it calculate weights and shipping? Generate a confirmation email when an order gets placed? Does it get built in a similar way that I'm working with WordPress in that you can see the item you are loading, preview the page, attach appropriate tags, etc? Is there a website that you know of that is using WordPress for their commerce that we can view?
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On its own WordPress does not handle e-commerce, you need a third-party e-commerce plugin. There are a number out there, each with their own feature set-
These are all great plugins (I'm partial to Cart66 and WooCommerce), and will do everything you've mentioned above except the CounterPoint integration. However, whichever Wordpress E-Commerce platform/plugin you use, make sure you are PCI Compliant.
All of these plugins will allow you to use either Paypal or some gateway's hosted checkout page which places you in compliance but weakens the brand and discomforts the potential buyer (something you don't want to do right before they're about to give you their money). You can either go the Paypal route, implement the PCI standards yourself (read: $$$), or use a hosted checkout page that offloads the PCI compliance. Check out this article about the complexities of PCI Compliance. Here's a snippet explaining the levels of PCI Compliance:
Credit card companies and merchants are cracking down on PCI non-compliance and the penalties are hefty. Here's the list of requirements on the PCI Security Standards website.
Here is a showcase of sites running Cart66.