This does not happen in a standard SmithCart environment but I did notice that your tabs use custom css as you can see the tabs flicker on mouse over.

1. On page load, tab1 style="display: block" but after modifying the quantity it changes to "display: none"

2. Notice that if you click on the quantity field, enter a number. then click on a tab it will display. However if you click on quantity then press tab, then click on a tab is does not display. Lost focus works differently for these similar events.

3. Lastly, click on the minimize icon in the top right corner of the module. Then click maximize and notice how the page is not displayed properly.

There must be something in your custom css or skin that is overriding the  behaviour. You might want to use the default skin to quickly test this.

Sounds like it is time to get around to it, or hire someone to do it for you.

First make a backup of your data using SSMS or SQLCMD into App_Data (preventing the file from being served by browser requests). Then copy the entire dnn web application folder (or export the application with IIS 7/7.5) to your development machine. Restore the data locally and start testing.

I use an automated script using winzip and ftp to syncchronize my development machine to my production server. It takes less than 5 minutes to start debugging DNN locally with an exact copy of my production website (2 GB) and data (250 MB).

One way to examine css is to press F12 in Internet Explorer. You can adjust the css and see the results immediately.

You can also run your css files using http://csslint.net/ although you may find the results overwhelming.