Likely to be the adaptor cable if it's cheap - might not be able to handle the HD throughput (although outputting to HDMI, you'd think it'd be able to).
While this sounds counter-intuitive, try moving around the alignment of the second screen as this might ditch the 'snow'.
Also, have you tried the screen with any other HD inputs?
Will the video from the MacBook to the monitor be as "computer perfect" through the HDMI in as it would though VGA or DVI inputs? Or will it be some kind of different reformatted "home theatre" image?
This question, as it's worded, is difficult to answer.
Quantitatively speaking, that monitor you're looking at has a native resolution of 1920 x 1080 pixels. Your MacBook can handle that and you will see a 1920 x 1080 pixel desktop on that monitor. Whether that's "computer perfect", given the term isn't something in the common lexicon, is up to you. My personal opinion is that 1920 x 1080 at the working distance most people sit at when they're at their desks, on a 27" screen, is going to look at little pixelated. Especially coming from a 1440 x 900 native resolution screen that's only 15" across that you have on a 15" MacBook Pro.
When it's all set up, will I just be able to sit at the desk, turn on the laptop, plug in mini Display Port cable to the Thunderbolt output, close the lid on the laptop and start typing away?
Yes. There's a small delay as the keyboard and mouse pair with the BlueTooth on the MacBook pro but it's usually less than 30 seconds.
Or even worse, will I have to keep the laptop lid open?
Keeping the lid open is not required but bear in mind that the MacBook Pro venting doesn't function well when the lid is closed. The cooling vents operate best when the lid is open. If you're running CPU-intensive applications you may find you need to operate with the lid open just to cool the computer adequately.
I'm not sure if the monitor has built-in speakers, but if it does, will the computer's audio come out of the monitor's speakers through the Thunderbolt-to-HDMI cable?
That cable passes audio, so yes.
Best Answer
From what I can tell from reading the Specifications Page, there are no speakers built into this particular monitor. There are however, two different audio outputs: HDMI 1.3 and 3.5mm headphone/speaker jack
The easiest way to test this out is just get a cheap set of speakers or a headset with the 3.5mm jack and plug them in. If it works, you know all is well with your setup - you just need speakers.