"Typically" the image tags are used to track impressions and your affiliate id is passed through the hyperlink itself, so you might be ok w/ the setup you have. I'm not familiar w/ how Amazon's program works, so I can't be 100% sure this is tracking properly. Did you ask them?
You can also see if the affiliate cookies are being dropped when you visit through the links you listed.
The 1x1 pixels are impression tracking tags and you can typically remove them. All the necessary tracking is accomplished with the click url.
General rule of thumb, if you see clicks showing up in stats for any given link, you're ok.
That said, most networks have rules and policies against modifying the link code. They won't bat an eye at it, unless you complain that their links aren't tracking. If you break something, it's your responsibility.