Yes, thank you!
The fix MPL Trace found worked, for the most part. The game now launches and it, so far, has fixed both the launch error and BattlEye error I was getting as well! Though I don't think it works 100% as intended because the opening screen is underwater and it keeps bringing up the EDEN popup whenever I start (Which may be normal, I'm not sure) I can at least play now.
Renaming 'api-ms-win-core-fibers-l1-1-1.dll' to 'api-ms-win-core-fibers-l1-1-1_bak.dll' is what fixed it for me.

I'm not fully sure if the RPTs use ISO 8601 or something else (YYYY/MM/DD vs YYYY/DD/MM), but the RPT on the far right has 2015 in it, so I'm assuming it's the one on the far left however if you could tell me how to make Arma make a new one I could make sure it's up to date and give it to you much more confidently.

Unfortunately, yes, I have checked out the other two tickets on the issue and neither seemed to help, though the bit that I don't understand is the RPT looking old since I've already reinstalled DirectX, VC++ Redist, GPU Drivers, and ARMA 3 itself probably two or three times each.

Odd, not sure why Mediafire is against Goo.gl terms of services.
Direct link is http://www.mediafire.com/download/9dzwohxv6b6j95z/ArmaReport_Log_20160609T012747.zip

List of "fixes" that I've tried: