Can I perform phone tasks through a Windows 7 PC directly? As an developer and QA tester, spending hours on my Xperia Z1 Compact isn't too cosy. Therefore, messing around with android emulation software like BlueStacks, I run into compatibility problems, mostly GAPIs (Graphic Application Program Interface). 

Therefore, I was wondering if there is a way to connect my phone to my Win7 PC with the USB cable, and have the phone do all the android processing, and all the PC will do, is enable USB keyboard and mouse as input devices for the phone, and an external monitor as the output device which the PC will boost the resolution from the phone into something that's a decent resolution on the monitor. 

But I don't know if the USB poll rate on the Z1 Compact is high enough to deliver all that information both ways without too much latencity. 

So, is this possible and is is already done?
What you want to achieve is nearly impossible.

You can get some approximations by using Airdroid app (lets you see your phone on the web browser from your desktop pc) or installing vnc server on your device and vnc client on your desktop pc, or using Pushbullet app. This last is used for notifications rather than absolute control, but you could give it a try