2 questions; Enough power for Mini ITX and case size.

Love the board here (first timer) but relatively surprised there isn't a power supply section. Onward!

WARNING: LONG READ!

So as the title says, I have two different questions. The first, is in regards to the power supply needed to power my parts list. I have used various online calculators but they all give different answers. What is the general consensus in regards to power?

Here are my parts list, excluding room for possible future upgrades.

Motherboard: GA-H77N-WIFI
Product: GIGABYTE GA-H77N-WIFI LGA 1155 Intel H77 HDMI SATA 6Gb/s USB 3.0 Mini ITX Intel
Link: http://www.newegg.com/Product/Product.aspx?Item=N82E16813128567

Processor: BX80637I53570K
Product: Intel Core i5-3570K Ivy Bridge 3.4GHz (3.8GHz Turbo) LGA 1155 77W Quad-Core Desktop Processor Intel HD Graphics 4000
Link: http://www.newegg.com/Product/Product.aspx?Item=N82E16819116504

RAM: CMZ8GX3M1A1600C10 (One stick for now, plan to upgrade to 2 later)
Product: CORSAIR Vengeance 8GB 240-Pin DDR3 SDRAM DDR3 1600 (PC3 12800) Desktop Memory
Link: http://www.newegg.com/Product/Product.aspx?Item=N82E16820233218

SSD: SDSSDX-120G-G25
Product: SanDisk Extreme SDSSDX-120G-G25 2.5" 120GB SATA III Internal Solid State Drive (SSD)
Link: http://www.newegg.com/Product/Product.aspx?Item=N82E16820171567

So now onto that, It's time to figure in a PSU... but first, let me explain what I need. Because I have already chosen my case, I'd really like to stick with this. I want an iDEA x2 case from ebay which is only 6.7in x 6.7in x 2.67in (6.8cm). That's really thin! (Remember this for later)

Case link: http://www.ebay.com/itm/251290205940?ssPageName=STRK:MEWAX:IT&_trksid=p3984.m1438.l2649

Obviously, the motherboard is about all that will fit. A clunky PSU definitely won't fit in there so I decided to go with an external one. The best I could do are the following parts:

PSU: ZVC220HD12S1
Product: Genuine Dell Optiplex GX620/ SX280 Ultra Small Form Factor (USFF) DA-2 220W AC Adapter
Link: http://www.amazon.com/Genuine-Dell-Optiplex-Included-Compatible/dp/B004578NQW/ref=wl_it_dp_o_pC_nS_nC?ie=UTF8&colid=1VEO4STPIGROV&coliid=I33I8AY1904NCM

PSU Module: ATX 250W DC Power Supply Module
Link: http://dx.com/p/atx-12v-250w-dc-power-supply-module-black-204019?utm_source=GoogleShoppingUS&utm_medium=CPC&utm_content=204019&utm_campaign=345&gclid=CK34iL_G-7cCFfBDMgodU2YAgw

Obviously the Dell PSU is the bottleneck of this whole operation which is the biggest one I can find and can only haul out about 220watt. I have no idea if this would be suitable enough to run the above parts list.

Problem #2: PSU wiring
Even if this combination could work, I honestly have no idea how to wire it up. I'm not afraid to get my hands dirty though.

Here's a forum post / how to guide to wire up a mini itx board but here the difference is that there isn't a 4 pin molex for the processor power in his guide, which we would need for our build.

Looking at the ATX 250 module, it DOES supply the 4 pin connector and the cable, so I suppose it would be just a matter of using the same wires for power?

Problem #3: The low profile case.
As I stated earlier, the inside of the case only sits at 2.67inches or 6.8cm.

The motherboard says 10 inches high but that can't be right. I'd figure in a few cm on the motherboard from the area of the processor and a few more cm for the spacing under the motherboard from the case.

Obviously the stock CPU heat sink cooler won't fit, standing at 3.2 inches (with processor?). But a low profile heatsink just might! Here, we have a Rosewill low profile CPU cooler that stands at just 2.6 inches. Whew, that's cutting it close! Will it fit?

Heatsink: RCX-Z775-LP
Product: Rosewill RCX-Z775-LP 80mm Sleeve Low Profile CPU Cooler - OEM
Link: http://www.newegg.com/Product/Product.aspx?Item=N82E16835200057

So that about sums up the problems with this current build. I apologize for the LONG read and hope you've stuck with me.

Any and ALL help is much appreciated.