It really depends what you need. There is a good breakdown of chipsets
here.
What is important to note is that chipset specs don't guarantee motherboard specs. By that I mean just because a Z87 motherboard is supposed to have X-feature, it might not if the manufacturers have cut it down to get it to a certain price point.
It's quite likely a H81 chipset is plenty. Some of it's limitations vanish at ITX level due to limitations of the format anyway. It won't run the CPU or graphics card any slower, so if it gives you more money to spend on performance parts then it's a great solution. The main downsides are only two SATA 6GBps (board dependant) and two USB3.0 (again, board dependant). It's not going to limit you as such, but a lack of connectors can be problematic.
If you want to keep the price down but increase the number of connectors, have a look at B85s. There aren't many at ITX format, but they are a decent option.
If you need SSD caching or integrated Wifi, you are basically forced to step up to H87, and you get various other features too.
Z87 is basically tied to two features - overclocking and full SLI support. At ITX, it means you can pretty easily ignore them if you don't need the former, as the latter is impossible. The exception to this would be if you needed an additional board feature on one of the higher end boards. The MSI and Asus high-end solutions both boast of improved audio, and all three of the Asus Z87 boards offer excellent CPU cooler compatibility if you are trying to transfer over an existing CPU cooler (though if you aren't overclocking, this shouldn't be necessary).
As for your build, I'd ignore the 4670. It's effectively the worst value for money i5 you can buy. A 4430/4440/4570 will perform near to identically, and give you some money to push into other places. By the time you get to the 4670, you might as well just get the K version, as it gives you more options and it's not really any more expensive.
At $1300 you can really do what you like as it's very flexible. Your starting point should probably be the case at ITX level, as it will be this which will impose limits on CPU cooler, graphics card and PSU size.
I'll refrain from build-spamming as I don't really know enough to narrow it down, but a $1300 gaming machine is probably going to be something along the lines of an i5, 8Gb of ram, 120Gb SSD, 1TB HDD, GTX780. A lot of that depends on what you need for your money.