First of all, your CPU is not THAT bad. It's actually a pretty good CPU (APU) with acceptable single core performance. There is no FM2+ CPU out there right now that you will experience a noticable boost in performance with. You shouldn't get a new CPU and mobo because you won't have much money left to spend on the graphics card so stay with the CPU (APU) you have and overclock it. If you still have the stock cooler, then pick up a 212 EVO or Cryorig H7 if you decide to overclock.
First of all, with that CPU you will be able to stream CS:GO and maybe BF if you put the settings down a bit, but I think it will be a problem streaming Far Cry 4 as it is a much more demanding game.
8GB of RAM is enough for most gaming and is good for streaming so you can spend most of your money on the GPU.
You will have to change the GPU and the PSU.
Because you have $400 to spend, you can really go high quality on this.
For GPU, get the awesome GTX 970, a solid card everywhere:
http://www.amazon.com/GeForce-Overclocked-256-bit-DisplayPort-Graphics/dp/B00NJ9BJ8G/ref=sr_1_4?s=pc&ie=UTF8&qid=1446950519&sr=1-4&keywords=gtx+970
The GTX 970 will be bottlenecked a bit by your CPU so that's why it would be good if you overclock your CPU.
Along with the top quality GTX 970, you need to pick up a top quality PSU:
http://www.amazon.com/EVGA-SuperNOVA-Crossfire-Warranty-110-B2-0750-VR/dp/B00KFAFRW6/ref=sr_1_1?ie=UTF8&qid=1446961488&sr=8-1&keywords=supernova+b2
This SuperNova B2 is also semi-modular so you can do some pretty good cable management.
That is pretty much all you need to know, sir. Cheers