I haven't personally used any of the services offered by EZDropship.com, but I did a quick research for you and I'll try my best to address each of your concerns:
EZDropship.com is one of the many dropping sites for Premier Products International. Premier Products Int'l s an eBay Certified Service Provider since 2006. They offer a wide range of quality products coupled with a fast and reliable service. Just for your reference, one of their dropshipping sites is currently listed as a verified supplier here on SaleHoo. Feel free to check them out by visiting this link: Link hidden: Login to view
The EZ Auction product ordering system is EZDropship.com's name for their dropshipping program. It actually works pretty much like most dropshipping programs found on other sites, where you register, then choose from the list of products they have, list it on your site/store and once you get a sale you process the order through there system and they'll take care of the rest. They do have a pretty hassle free system and do offer detailed inventory and stock level information, plus email alerts when out-of-stock items become available or when new products are about to arrive.
So far, I wasn't be to come across any negative feedbacks about their EZ auction ordering system.:)
The US$2 is a handling fee that is not included in but charged on top of the actual shipping cost. You will be billed for the actual shipping cost of an item. You may obtain a shipping rate for your customers by visiting the Product Page, clicking on the picture of whatever you are inquiring about, and a UPS window will appear allowing you to enter the zip code you may be shipping to.
Hope this helps! Cheers :)