Sata: while other cards have been shown to work most people stick with the via6421 cards as they are cheap and they are near certain to work. Most will come with an external port too.
I am however not entirely sure what you are concerned about with the direction of the slots as they can come facing any way save upwards and sata cables are nice and flexible.

Opening the 360 is easy enough. I don't even really use a poor mans opening tool but they are a few dollars and I am well versed in fixing plastic should I ever care to.
http://www.llamma.com/xbox360/repair/Xbox-...Disassembly.htm and http://www.xbox-modchips.com/xbox360-tutor...ening-guide.htm might be of interest and for less than $5 I think you can get a tool these days.
Still http://www.youtube.com/watch?v=qaxIB6cXTYM&fmt=18 is also good and it requires only basic tools (the most elaborate being the t12 torx)

Once you have the drive: the 360 uses an uncommon power cable (I do not want to say custom exactly) so people either use the 360, make their own (pinout here: http://forums.xbox-scene.com/index.php?showtopic=668247 ) or buy an adapter (something like a 360 xtractor).

Once you have the drive (including some way to power it) then you connect it to the sata, now is where it gets tricky though as there are all manner of things people like to do when powering the drive. SATA is more or less capable of hot unplugging but some people still like to power drives with their system, others like to power drives as they launch the flashing app, a short while before the flashing app and it goes on and on with varying methods and tools.
Generally your app is sending custom ID commands (why you need a card capable of forgoing things like the via 6421) for a good while before it gives up so I say power whenever you like.

 

Believe me I am familiar with the cramped case this *spits on big box vendors*. However most of the cards I see are shaped like this:
http://www.newegg.com/Product/ShowImage.as...0un-RAID%20mode
or with a connector at the back like on your example.
Link in case your browser plays up ( http://www.newegg.com/Product/Product.aspx...will-_-16132012 )

In theory it might work and some have claimed it but I have never seen it (aside from the newest liteon I have now seen and flashed them all and spent a great deal of time messing around with things). As for booting the software while I am a great fan of dosflash some of the newer tools can do it all from windows and do so very well (older stuff was a bit patchy at times for a lot of people).

I should have said before but a lot of the "external" sata ports are in fact eSATA (a nominally different, albeit compatible if you have an adapter cable, shaped connector).

 

esata and sata is just like the different shapes of USB albeit with slightly more rare adapter cables. Thanks for the pictures (for a close up shot that is awesome) as they confirm what I suspected and I should mention a most of the adapters I see will point towards the camera or towards the motherboard battery.

While I am thinking about it you can also get sata leads that have a 90 degree (or 270 depending on how you look at it) angle to them.

 

I watched the video and the software part is great. I will however offer some discussion on everything else:

You are only turning the xbox on and off to utilise the ability to power the drive and nothing more; theoretically the drive could be powered by the sata (if you look on the back it will mention 5V and 12V- 12V is for the motors and whatnot while 5V which the sata cable should be able to do is for the onboard logic) but as I said I have never seen it work that way.
There is no data flow between your PC and xbox or even the drive and the xbox when flashing it (your PC and xbox cases should be making an electrical connection though, do a forum search for floating ground if you want more).

Turning the 360 off and on initialises the drive which is good when the software is trying to hook things and initialise things or do things that only work on power up*.
The tray half open thing in the video is an odd one for the benQ (it is newer and more commonly associated with the liteon) but it is a perfectly valid method and that is how I would do it if I were to be doing it with only a 360 and no cut/jumped wires.
I think it mainly comes down to if you are not reading, erasing or writing and you are having trouble accessing the drive then power cycle, some like to power cycle at each "stage" in flashing and others are used to something like DOS which does not exactly multitask (one app for reading/flashing and another to make the hacked firmware) which means a power cycle for reasons above.
The only real way to screw it up and not be able to pull it back is to not have the key and then erase it/write something new which is why everyone will say backup the key (email it to yourself and others, USB drive, several copies about your hard drive and other machines in your house......) once you get it.

*most of these drives need to be put into another mode (usually something along the lines of recovery/debug) to be flashed. What you do when you have a drive half open (actually you close the drive as you normally would so it registers as closed but with the tray not all the way there as you have cut power before it is actually closed it sets it into this mode).
BenQ drives have several methods as you may have guessed and as you are seeing a lot of these guide makers border on superstition a lot of the time and others are somewhat opposed to learning something new "it works so why change it".

In conclusion you can use the 360, make your own device or as you say buy one of the somewhat expensive premade things (I do have one and they are nice but for anything other than a liteon or flashing many drives* I do not suggest it).
I will mention that some people suggested it might be a banning route (on power diagnose: nobody should ever not have the DVD drive plugged into both power and sata on the 360) but I doubt it very much.

*as you seem to have followed thus far you probably know most of what there is not know about flashing a 360 drive (or nothing 5 minutes reading can not fix) so it could well end up like the time you let on you can fix computers/build websites/operate basic software/similar and people came crawling out of the woodwork.