£1300 GBP Complete Gaming System Build

"Any help will be very gratefully received."

This is the most courteous thing I have even read on any forum in a decade. Anyway, let's get to solving your issue.

Taking into consideration your needs here's how I'd prioritize the CPU i5 -3570K > i5 - 4790K > i7 - 4930K

The reason being that 4th generation Haswell offers higher theoretical performance which, in real world, translates to a marginal 'perceived' improvement which you are never going to benefit from. Clock for clock and 4th generation chip is 6% more powerful than a 3rd generation one, which adds a net total of 250 MHz extra power to your system. The most processing intensive entity on your list (gaming) doesn't use that much power and CPU is almost never a bottleneck in any gaming rig. In fact, there are very few games that are capable enough to use the quad core processor. Most are optimized only to use two cores. So the 'on paper' benefit that a 4th gen processor has over a 3rd gen never translates into any real world advantage, not to mention the extra cost.

On the contrary Haswell brings its own negative aspects to the mix and has a net negative impact. Like higher operating temperatures and lower overclocking potential. The 4th gen chips also have superior onboard graphics compared to 3rd gen, again, a feature which is redundant to you given that your dedicated GPU will bypass it, but you will still end up paying a premium for it anyway. Basically, a 4th gen chip costs you more and brings no value to the table. They are more suited for Laptops and portable devices. The 3rd gen chips also overclock better than the 4th gen chips and are more sturdy (resilient) in the face of high operating temperatures.

It's the same thing with i7. It is relatively more powerful but adds no practical value to your setup since it is unlikely that your system will ever stretch to its limits to ever use it. My verdict : go with the i5 - 3570K. It's a good combination of power, stability and value. Beyond a certain point, the law of diminishing returns kicks in and a cutting edge CPU means nothing more than a higher price/performance ratio. The i7 is recommended for people who work with applications which rely on raw power and have the capacity to leverage multicore technology, like Photoshop and 3D modeling engineers, animation specialists. 'Better future compatibility' is a farce to scare people into buying things they don't need.

More often than not (read : always) the GPU is the bottleneck in a gaming PC. Allocate a higher % of your budget to this. A one piece GPU is preferred over dual GPU SLI for compatibility reasons and especially for your 'quiet and cool' requirement. Some games that are incompatible with SLI end up giving poorer performance than a single card setup.

Cooling and noise : Go for a custom water cooling loop. This will reduce the number of intake fans and as a result keep your system from sucking in less air and consequently, less cat hair. It will also be cooler than the average air cooled PC.

Hard disk : Opt for a (256 GB SSD for OS and games + 2 TB Mechanical Hard disk combo) instead of a single 3TB HDD. The former makes the system faster in general and games load faster.

The 32 GB ram is an overkill if you aren't going to be creating a RAMDisk system or leveraging a quad channel RAM. 1866 (8GB X 2) dual channel RAM works best for you.

Case : I think, your interests will be better served if you mod the case instead of using a high end ready-made case. Use a bare bones roomy case as a template and build on it. Add a custom water loop, add rubber grommet mounts to fan mounts. OR you can commission a carpenter and build the PC inside the desk. This will keep the system constantly elevated, hence less cat hair, and it will also provide a good sound buffer. Check these

http://www.geek.com/wp-content/uploads/2012/12/3nov6.jpg

http://www.sweclockers.com/image/red/2013/08/26/Redharbringer_0007_Layer+3.jpg?t=original&k=c329a166