Computer A: It is at least $40 more than listed, because it doesn't show the "Case" price in the list that you have generated.
1) It should automatically detect the GPU and use it for your graphics needs.
2) Yes the GPU will fit well, as long as you don't have the immediate slot below the 1st PCI-e (16x) slot occupied (that is the PCI x1 slot - small slot). By the way, the Mobo supports 2 x PCI-e GPU's. One is at 16x and the other 4x speed.
3) Yes you can fit a CPU cooler in, you just have to be a little selective on which ones. Most will work, you'll just have to make sure that the one that you get doesn't get too wide to interfere with the GPU area. Also your RAM selected should be okay for most CPU HSF's. It has what I'd call a "Medium" profile heat spreader, meaning that the heat spreader protrudes upwards a bit from the RAM module itself. I prefer to get "Low Profile" heat spreaders, so you have all of your options open.
4) Yes SSD's are TOTALLY good to go! All you need is SATA power and a SATA cable and an open SATA port on your mobo and your are good to go! The Fractal Design Core 1000 says that it has room for 3 x 2 1/2" drives (SSD's), so you should be good there too. You can even use double sided tape to attach the SSD to the wall of the case, if you are so inclined. Since SSD's have NO moving parts, you can pretty much just throw it in the case and your good (although I'd try and make sure it's not hindering airflow).
Computer B:
1) That GPU is okay, I'd rather have the Sapphire one, but it's $15 more, so that is up to you (link below).
http://www.newegg.com/Product/Product.aspx?Item=N82E16814202011
2) You can fit a CPU cooler in later (if you so desire), just the same requirements as stated for Computer A above.
3) The FX 4100 is a decent CPU for the $. If you can afford the FX 6300 that would be better (6 core) and probably last a bit longer than the FX 4100.
4) SSD's are always good for the future. Same requirements as from Computer A above.
5) Sound cards are fine, but unless you really need it, most onboard sound is quite good these days. I'd personally use the $ for the sound card on other upgrades, IMHO.
6) Yes you can fit everything into the case.
Let me make you a build like yours and you can see your options:
Computer A:
[url=http://pcpartpicker.com/p/TvJ1]PCPartPicker part list[/url] / [url=http://pcpartpicker.com/p/TvJ1/by_merchant/]Price breakdown by merchant[/url] / [url=http://pcpartpicker.com/p/TvJ1/benchmarks/]Benchmarks[/url]
[b]CPU:[/b] [url=http://pcpartpicker.com/part/amd-cpu-ad580kwohjbox]AMD A10-5800K 3.8GHz Quad-Core Processor[/url] ($128.79 @ Amazon)
[b]Thermal Compound:[/b] [url=http://pcpartpicker.com/part/arctic-cooling-thermal-paste-mx2r]Arctic Cooling MX-2 4g Thermal Paste[/url] ($7.99 @ Newegg)
[b]Motherboard:[/b] [url=http://pcpartpicker.com/part/asrock-motherboard-fm2a75pro4m]ASRock FM2A75 Pro4-M Micro ATX FM2 Motherboard[/url] ($78.18 @ Newegg)
[b]Memory:[/b] [url=http://pcpartpicker.com/part/gskill-memory-f314900cl9d8gbsr]G.Skill Sniper Series 8GB (2 x 4GB) DDR3-1866 Memory[/url] ($60.30 @ Newegg)
[b]Storage:[/b] [url=http://pcpartpicker.com/part/western-digital-internal-hard-drive-wd10ezex]Western Digital Caviar Blue 1TB 3.5" 7200RPM Internal Hard Drive[/url] ($66.64 @ Amazon)
[b]Case:[/b] [url=http://pcpartpicker.com/part/corsair-case-300r]Corsair 300R ATX Mid Tower Case[/url] ($49.99 @ Newegg)
[b]Power Supply:[/b] [url=http://pcpartpicker.com/part/corsair-power-supply-cx430m]Corsair CX 430W 80 PLUS Bronze Certified ATX12V Power Supply[/url] ($39.99 @ Newegg)
[b]Optical Drive:[/b] [url=http://pcpartpicker.com/part/lite-on-optical-drive-ihas124-04]Lite-On iHAS124-04 DVD/CD Writer[/url] ($17.99 @ Newegg)
[b]Operating System:[/b] [url=http://pcpartpicker.com/part/microsoft-os-wn700404]Microsoft Windows 8 (OEM) (64-bit)[/url] ($94.04 @ NCIX US)
[b]Keyboard:[/b] [url=http://pcpartpicker.com/part/rosewill-keyboard-rikb11002]Rosewill RIKB-11002 Wired Slim Keyboard[/url] ($12.47 @ Amazon)
[b]Mouse:[/b] [url=http://pcpartpicker.com/part/gigabyte-mouse-gmeco600]Gigabyte GM-ECO600 Wireless Laser Mouse[/url] ($24.99 @ Amazon)
[b]Total:[/b] $581.37
[i](Prices include shipping, taxes, and discounts when available.)[/i]
[i](Generated by PCPartPicker 2013-04-28 00:53 EDT-0400)[/i]
* Changed HD, Case, & PSU to match up with build. Removed sound card too, but that is up to you.
Computer B:
[url=http://pcpartpicker.com/p/TvNk]PCPartPicker part list[/url] / [url=http://pcpartpicker.com/p/TvNk/by_merchant/]Price breakdown by merchant[/url] / [url=http://pcpartpicker.com/p/TvNk/benchmarks/]Benchmarks[/url]
[b]CPU:[/b] [url=http://pcpartpicker.com/part/amd-cpu-fd6300wmhkbox]AMD FX-6300 3.5GHz 6-Core Processor[/url] ($135.34 @ Amazon)
[b]Thermal Compound:[/b] [url=http://pcpartpicker.com/part/arctic-cooling-thermal-paste-mx2r]Arctic Cooling MX-2 4g Thermal Paste[/url] ($7.99 @ Newegg)
[b]Motherboard:[/b] [url=http://pcpartpicker.com/part/asus-motherboard-m5a97r20]Asus M5A97 R2.0 ATX AM3+ Motherboard[/url] ($84.99 @ Amazon)
[b]Memory:[/b] [url=http://pcpartpicker.com/part/gskill-memory-f314900cl9d8gbsr]G.Skill Sniper Series 8GB (2 x 4GB) DDR3-1866 Memory[/url] ($60.30 @ Newegg)
[b]Storage:[/b] [url=http://pcpartpicker.com/part/seagate-internal-hard-drive-st31000524as]Seagate Barracuda 1TB 3.5" 7200RPM Internal Hard Drive[/url] ($67.99 @ NCIX US)
[b]Video Card:[/b] [url=http://pcpartpicker.com/part/sapphire-video-card-100358l]Sapphire Radeon HD 7770 GHz Edition 1GB Video Card[/url] ($99.99 @ Newegg)
[b]Case:[/b] [url=http://pcpartpicker.com/part/fractal-design-case-fdcacore3000usb3bl]Fractal Design Core 3000 USB 3.0 (Black) ATX Mid Tower Case[/url] ($54.99 @ Newegg)
[b]Power Supply:[/b] [url=http://pcpartpicker.com/part/corsair-power-supply-cx500]Corsair Builder 500W 80 PLUS Bronze Certified ATX12V Power Supply[/url] ($44.99 @ Newegg)
[b]Optical Drive:[/b] [url=http://pcpartpicker.com/part/lite-on-optical-drive-ihas124-04]Lite-On iHAS124-04 DVD/CD Writer[/url] ($17.99 @ Newegg)
[b]Operating System:[/b] [url=http://pcpartpicker.com/part/microsoft-os-gfc02050]Microsoft Windows 7 Home Premium SP1 (OEM) (64-bit)[/url] ($90.91 @ Amazon)
[b]Keyboard:[/b] [url=http://pcpartpicker.com/part/rosewill-keyboard-rikb11001]Rosewill RIKB-11001 Wired Slim Keyboard[/url] ($9.99 @ Amazon)
[b]Mouse:[/b] [url=http://pcpartpicker.com/part/gigabyte-mouse-gmeco600]Gigabyte GM-ECO600 Wireless Laser Mouse[/url] ($24.99 @ Amazon)
[b]Total:[/b] $700.46
[i](Prices include shipping, taxes, and discounts when available.)[/i]
[i](Generated by PCPartPicker 2013-04-28 01:00 EDT-0400)[/i]
* Changed CPU, Mobo, RAM (low profile heat spreaders), & GPU