How well does the new iPad work with the iPad Dock that Apple sells? I am wondering if the increase in weight and thickness throws off the dock connector or the alignment / balance of the iPad dock connector.

I am asking about the part MC940ZM/A which is listed as the iPad 2 dock on the bottom sticker - but everywhere else says iPad dock without reference to which generation.
Macworld run a first test with a collection of accessories:

In our testing, even many form-fitting iPad 2 accessories work with the third-generation iPad. For example, most dock-cradle accessories—such as speaker docks, chargers, and even Apple's iPad Keyboard Dock—released prior to the new iPad work fine. (This is likely because many of these accessories were designed to fit both the iPad 2 and the original iPad, and the original iPad is even thicker than the new model.)
Similarly, most iPad stands and mounts—especially those that hold the iPad loosely in a groove, slot, or cradle, or on easel-style supports—work perfectly with the iPad. Even Twelve South's HoverBar, which uses a rigid-plastic, corner-hugging cradle, fits the new iPad, albeit with a tighter fit than when mounting an iPad 2.

Even better, Apple's site now states that the MC360ZM/A dock lists compatibility with the iPad 1st generation and the MC940ZM/A dock with the iPad 2 2nd generation and iPad 3rd generation.