is the intel core i7 6700 good for 1080p live streaming?

More likely you would get the boost you need by replacing the HDD with a SSD, and perhaps bumping the memory up. The big difference between a 6400 and a 6700 is in number of threads, the i5 allows half the concurrent threads that the i7 allows.

That depends on the software. Most applications don't use very many threads, because it's a pain to program well for multiple threads such that they don't get in each others way. The OS allocated threads such that as many things can happen at once as possible, but if you are only running a few apps you aren't likely to need more threads. Try running Resource Monitor and watch the CPU usage and individual cores activity to try and get a handle on what's being used. Next check the memory usage load and contrast that with your physical memory, if you are going over the physical then it'll swap out and really slow you down.