I have not done much live streaming lately, but I have used the Blackmagic Intensity Pro and Intensity Thunderbolt products for the purpose. The trouble with Blackmagic stuff is that it is pretty particular about input formats over HDMI and it takes a bit of trial and error to get it 100% depending on the camera you use.
What type of live streaming are you wanting to do? That is really what will dictate your needs for everything you've asked for advice about. If you just need to stream video from a single source, you can pretty much pull that off using any Mac with a 2nd Gen Intel Core Processor or newer. If you need to be able to use your Mac for other tasks during the stream, or stream 60 frames per second, then you'll want to make sure you have at least an i5 and preferably an i7 processor from the last couple of years. There are hardware video encoders out there, even ones built into modern CPUs and GPUs, however, unless you want to spend a fair bit of extra money on some specialized hardware, the quality of the encoding is not on par with what you get using a solid software (CPU based) encoder in most cases. So you don't necessarily need a high end GPU unless you're planning on streaming games or other software that need the GPU horse power.
It surprisingly does not take a huge amount of power to pull off a basic HD live stream. My wife occasionally likes to stream a couple of non demanding 2D games on Twitch. She uses a 2nd Gen Core i3 based Hackintosh with 8GB of RAM and an nVidia 740. Live encoding a 30fps 720p stream and playing the games pretty much maxes out that little box. It is able to pull it off smoothly but there is zero overhead for anything else.
On the other end of the spectrum, at work I have a 2012 i7 MacMini quad with 16 GB of RAM and the Blackmagic Intensity Thunderbolt. We use it to live stream meetings and trainings and it has plenty of power to stream 1080p from a camera source and desktop capture and also be able to do just about anything else at the same time. Love that little box.
