ASUS is known to do some custom stuff, for instance they have their own version of Realtek's ALC1220 audio, which they dub s1220a.
I don't get why you insist on clinging to the H series MBs though, especially if you're limited on funds. ASUS MBs tend to cost a bit more.
Anyways, you've yet to mention a budget, but as I already stated, a GTX 1060 would be quite a step up, even on your current MB. Take care of the GPU first. If you can at some point afford an overclockable MB, then do that later. You'll need an aftermarket cooler to OC the CPU though. If you ask me, such things are not top priority on a budget build, especially since you already have a pretty good CPU, even at stock speed.
I can't stress enough that the real weak link in your rig is definitely the GPU, not the stock CPU speed. Even the non OCable MB is practically a non issue compared to your GPU.
As for why Crysis 3 runs poorly, it's a combination of advanced shaders, a poorly optimized engine, and their having tied together certain settings, which makes it impossible to fine tune settings. You pretty much have to drop shaders and shadows pretty low on weak spec.
That said, I really liked that they at least offered tons of script commands on Crysis 1 so users could really fine tune settings beyond what you could do in the game's graphics settings. Because of that I was able to come up with my own custom tweaks on Cry 1, which allowed me to run it on a mere P4 single core CPU, with mediocre GPU, with max textures, med shadows and shaders, at 1200x900, with plenty playabe frame rate.
The reason they stopped offering script commands in Cry 2 and 3 was because too many players whom didn't have a clue how to use them complained that the game was horribly unoptimized, so CryTek over reacted and dumbed down the settings on Cry 2 & 3. So I blame the noobs just as much as CryTek for the demise of Crysis.
For the record, I have no experience in programming or even extensive modding. All I did was read the TweakGuides.com guide for Crysis 1 until I found which commands to use. Other than that I spent about an hour of trial and error script changes until I found out which command to use to keep the foreground from being blurry. It was a combination of using Low Object Quality, with High veg draw distance.