I did a little bit of testing and found that it's skipping steps when checking the Network through the Chrome debugging tools.
The working process (the Abantecart demo) shows these steps here:
http://demo.abantecart.com/index.php?rt=sale/order/update&order_id=17&product_id=118&s=demo_admin&token=mVxaFejkCmYO5Npx0o7ffXv9gipMD9HI
http://demo.abantecart.com/index.php?rt=sale/order/recalc&order_id=17&s=demo_admin&token=mVxaFejkCmYO5Npx0o7ffXv9gipMD9HI
http://demo.abantecart.com/index.php?rt=sale/order/details&order_id=17&s=demo_admin&token=mVxaFejkCmYO5Npx0o7ffXv9gipMD9HI
And then it simply goes on through a bunch of JavaScript and CSS requests.
While the results for the not working process:
https://3cdog.net/index.php?rt=sale/order/update&order_id=628&product_id=1490&s=(details removed)
https://3cdog.net/index.php?rt=sale/order&s=(details removed)
And then it does the rest of the CSS and JavaScript requests.
Again, there are no errors in the logs or anywhere else that initiate when this happens.
Based off of this info, are there any files you recommend I look at?
NOTE: I also replaced the entire core directory and the admin directory with the newest files from the GitHub repository.
Maybe it's a database problem?
Another specification, changing any of the values from the textboxes works fine, but anything that opens up a window with the product details doesn't work. The two screenshots below show what I mean (Capture.PNG & Capture1.PNG).