If you were going to switch to intel, you might be better off with an i3-6100 . With its stock speed, it's stronger than the older i5 in single thread apps and the HyperThreading takes advantage of that too, so in 3-4 thread apps it'll run close to the same as the i5. The only advantage the i5 holds is in full load apps where the single thread/core has greater throughput than a shared thread/core. But unless benchmarking or rendering video, chances of running all 4 threads at 100% or close are pretty slim. There's also upgraded ram, better audio, better usb/sata support, better compatability with dx12 and other latest Gen software. And that's not to mention the upgrade possibilities. The only step up for the i5 is an i7 or Xeon v3 which is only of use to a select few apps. The i3 can be upgraded to an i5 or i7 at a later date if performance becomes an issue.
The as rock pro is ok, not great, but ok and the 200r is a respectable budget case with plenty of possibilities. Not bad choices, but there are better. It might be worth looking into selling the old pc whole and working, you'll get more money than parting out the older stuff.
If you haven't oc'd the 6300, you might consider bumping it up to 4.0+GHz with a decent cooler and dropping some cash on a bigger gpu. You'll get more performance there than switching to intel as far as games go, but if you actually use the pc for a pc, not a substitute game macbine, then the move to intel could be worth it depending on what software you use most frequently