Does this address the app gap?
Yes and no.
You'll get to run x86 applications, but the majority of these aren't "phone friendly" so it's not really going to address the "app gap" when we talk about applications available for a phone. In terms of tablets, majority of applications won't be "touch friendly" too, but this would be no different to the current Surface pro line i.e. fairly insignificant and bearable.
It will probably be better on the tablet side of things (i.e. less of an app gap because of emulation), but probably not for phones since the application ranges and usages are quite distant. And desktops never had an app gap to begin with.
Assuming it's all emulated in software, the performance degradation will also mean native ARM counterparts will perform better, and some x86 won't emulate at a usable level.
Does this mean apps will be written for ARM and x86?
Applications targeted for ARM architectures should continue being written for that architecture. x86 applications will still be written aimed towards x86. Some languages and toolsets allow targeting multiple platforms from one code base, in which case, they would have two compilations (arm and x86) and need not make use of emulation.
They would not be mixing x86 code with ARM code. In theory, they could if they needed to use a 3rd party library written elsewhere, but the idea here is to make applications only available on x86 architectures working for ARM, not for developers to create x86 targeted specifically for ARM.
We may see developers create x86 applications that know they are being emulated on ARM (WOW64 allows this) and maybe alter how they function (make phone orientation friendly, for example) but since most programming solutions allow for compilation into both x86 and ARM, somewhat pointless.
Will phones be running an emulator for x86?
I'm inclined to believe it's software emulation so yes. Others are inclined to say there's some hardware involved, and so no to a varying degree.
Basically, there's no solid confirmation on how it's done yet.