Yeah, that's exactly how it works. The processor doesn't have to come out, it's just a matter of cleaning off the old thermal paste, applying new paste (usually comes with new CPU coolers) and replacing the cooler on top of the processor.
Depending on the design of the cooler and your case, you might have to take the motherboard out to install a backpack on the underside for the CPU mount. Not too difficult.
The power supply seems fine, 650 watts is more than enough for the flagship Nvidia GTX 980 if you ever wanted to get that. Definitely reusable.
The motherboard can be reused. But it'll tie into what kind of case you will be getting, as well as the cooler. Questions below.
No problem.
I just realized that I forgot to mention some things and ask you more questions. The Witcher 3 doesn't have official PC requirements yet,
but it seems likely that the 560 Ti will run it on medium settings.
If you want to play Witcher 3 and Ground Zeroes on high end settings, then a graphics card upgrade is in order. However, it's not that bad an idea to wait until the games come out (or at least wait for PC requirements reveal) and then decide if you want an upgrade. Black Friday sales aren't always the best time of year to upgrade, and by the time Witcher 3 releases, the new Nvidia GTX 960 should be close to release. That would be an ideal upgrade for less than $300.
Now, the case, motherboard, and cooler. You said you wanted a small console-sized case. Unfortunately, it'll have to be bigger than that if you want to reuse your current motherboard. I don't recommend getting a smaller motherboard just for a smaller case, it'll be a waste of money to buy 6 year old motherboard technology, even if you buy a used one (used motherboard quality is subject to weardown and damage anyway).
There are small cases that I can recommend, as well as a decent water cooler for the processor that will fit in compact cases and allow you to overclock the processor. When you said you wanted it console sized, is it because you are planning to put it into the living room? There are home theatre style cases that look nice and not out of place in an entertainment setup. Or would a normal tower case be alright?