Hi, AJ - and welcome to TPU.

My advice would be as follows:
This is very outdated hardware, as you may have noticed from the low price tag. Any modern i5 CPU will outperform both of those E5405s, although they provide more cores. Those are very "entry-level" DP Xeons without HT at only 2 GHz, so to comment on this:
I don't know how old that i7 that you compared it to was, but any recent i7 (or maybe even i5) will beat the cr@p out of those E5405s
PSU might be OK, I had troubles running my two 80W TDP E5620s before even with a 750W PSU though. Your case must be big enough to fit this SSI-CEB form factor board. As long as it fits in there you'll be fine, you may have to drill some extra holes, though. As for RAM, this particular board only supports fully buffered RDIMMs, so you won't be able to use your old RAM in there. As for the GPU, this board will limit bandwidth on the PCIe x16 slot to 8 lanes. Should not be much of an issue with your graphics card, though.
I totally agree!

Why not indeed. It's just an unconventional way to do so. So what.
No, not at all. What is important to understand is that newer hardware always will beat outdated CPUs/MBs. Any mid-range barebone for, lets say 400 bucks will be way more powerful, have the latest technology and IMO would be the better solution overall.
As you already have purchased that setup, you might as well give it a try and see if it runs your game (h1z1 - never heard of that one)

. Of course you can re-use your disk drives and such. Just keep in mind that hardware is very old and you should definitely not update it (buying expensive DDR2-RAM, for example.) Another option would be to upgrade to a new mid-range i5 setup or the like - you'll notice the difference right away!

There is one other option... if you should be interested. I've got components that I no longer need (after having updated my system) now rotting in my basement and I don't really care to sell them, as I don't know anyone who could use that stuff:
- 2 Xeon E5620 CPUs, 4C/8T @ 2.4 GHz, LGA1366 "Westmere-EP", used for 1 year.
- 32 GB KVR DDR-1333 RDIMMs (4x8 GB), also used for 1 year.
Seeing as you could use that stuff, I would like to sell these to you, if you're interested for a very LOW price

All you would need to get is a DP 1366-socket board, they are hard to get these days, you can check out some of those boards
HERE at SuperMicro. If you should be interested, just ring the bell.

Hope I was able to help you.