defiantly, for streaming, the i7 3770 does far far better then all i5's, including your 2500. A GTX 770 (2GB) is beast even for 1080p, and for now exceptional for 720p as it will max out anything getting really high FPS.
With the GTX 770, Going with the 4GB version is pointless just a marketing gimmick.
For starters, The GTX 770 has a 256-bit memory bus which is designed for 2GB of Vram, not 4 so it is actually being limited and will never use the other 2GB. In games both the 2GB version and 4GB perform exactly the same with no advantages whatsoever by the 4GB.
Only advantage of the 4Gb version is if you have multiple screens, and even that there is only a small benefit as the memory bus is only meant for 2GB, and 128 Bit is meant...