Both are excellent choices but if you think a few frames per second makes that much of a difference and price doesn't really matter, get the i7-3770K because you can overclock a bit more easily because of the higher clock rate. However, both are indeed quad-core CPUs, so general applications will feel very similar to each other. But looking at the gaming section of this http://www.bittech.net/hardware/cpus/2012/05/01/intel-c...
a highly overclocked i5-3570K can actually beat the i7-3770K. I am not completely abidding by this website's benchmarks but as far as I'm concerned there is little difference and the best bet in your case is the i7-3770K.
as for a Z77 mobo,
you can get two options if SLi is your thing:
ASRock Z77 Extreme6 LGA 1155 Intel Z77 HDMI SATA 6Gb/s USB 3.0 ATX Intel Motherboard $139.99
This one offers a sweet spot between quality and performance and quite frankly, it has all the ports you need. This has a bit more PCI-lanes (not PCI 3.0 x16 lanes as both these two have two)
BIOSTAR TZ77XE3 LGA 1155 Intel Z77 HDMI SATA 6Gb/s USB 3.0 ATX Intel Motherboard with UEFI BIOS
This one is very similar to the one before, only that it's cheaper and priced at $121.99
There is a $169.99 variation of the first one (ASRock Z77 Extreme6 LGA 1155 Intel Z77 HDMI SATA 6Gb/s USB 3.0 ATX Intel Motherboard), however, the features are extremely similar and you are better off getting the cheaper version.
One step down is:
BIOSTAR TZ77B LGA 1155 Intel Z77 HDMI SATA 6Gb/s USB 3.0 ATX Intel Motherboard with UEFI BIOS $99.99
The main thing about this is the presence of only one PCI 3.0 x16 lane, which renders SLi unfeasable unless GTX 690 is your type of thing.
If it's just games, you want the i5. The i7 has no performance advantage in games. However, it comes into its own with applications that are heavily threaded and CPU limited. Examples are photo/video editing, video transcoding, modelling and simulation.