Hello,
the issue will be most likely caused by corrupted .exe or .dll file, but it might be a system file that is corrupted.

Could you please download Microsoft Dependency Walker - x86 version and open the Arma 3 executable with it, then save the report from it (File/Save), zip the .dwi file and upload it here? It will show all the libraries game is using and also reveal the corrupted one.

Ok I uploaded it

Hello,
according to the file you've uploaded, the library that is corrupted is "C:\Windows\SysWOW64\d3dx10_43.dll". This is a DirectX library, so I'd suggest to delete that file manually and then re-install the DirectX runtime from https://www.microsoft.com/en-us/download/details.aspx?id=8109 . (Please note that it's important to delete the file before the re-installation, because otherwise the file may not be replaced :/)

It asked me to choose the location I wanted to place the extracted files so I chose C:\Windows\SysWOW64

But it said the folder is invalid. Please make sure the folder exists and is writable

I'm sorry, I've sent you the wrong link; the one I wanted to sent you seems to be dead at moment. But that doesn't matter as you can use the file you have downloaded.

Just extract the files to a new folder, for example on your desktop. In that folder there will be a DirectX installer (DXSETUP.exe), so run that.

Ok I did what you said then ran ARMA and got this error instead
0xC0000135 - STATUS_DLL_NOT_FOUND

Hello, The OS is unable to load a file that is required to run the game. A missing DLL may belong to the game, the OS or one of the required components (DirectX e.g.). Possible solutions:

Verify the integrity of the game cache using Steam.
Re-install DirectX.
Uninstall the Visual C++ 2013 Redistributable package (x86 version), restart your computer and install the package again (do not use the Repair function).
Run the Windows System File Check tool to repair corrupted system files.

Hi Alex,
could you run x86 Dependency Walker again to find out what library is causing problem this time?

Try to install OpenAL, for example from here http://support.focus-home.fr/oalinst.exe

Ok I posted the x86 dependency walker status and I downloaded OpenAL with no problems

Here is the full ARMA 3 report as well

Arma 3 Troubleshooting Report

Sorry for the inconvenience.
Please report this on Arma 3 Feedback Tracker (http://feedback.arma3.com). Thanks to you we may fix the issue faster.

----------------------

Process exit

Exit code: 0xC0000135 - STATUS_DLL_NOT_FOUND
Running time: 00:00:02.7275750

----------------------

System information

Date: 2015-11-02 21:27:52 / 2015-11-02 15:27:52 -06
Current dir: D:\SteamLibrary\steamapps\common\Arma 3
Command line: "D:\SteamLibrary\steamapps\common\Arma 3\arma3launcher.exe"
CLR version: 4.0.30319.42000
OS Version: 10.0.10240.0 (Microsoft Windows NT 10.0.10240.0)
OS 64 bit: True
Process 64 bit: False
Culture: en-US
UI Culture: en-US

----------------------

Last game report

not provided

It also appears all the files with -WIN- Are the only ones that say Error opening file. The system cannot find the file specified(2)
With the excpetion of IESHIMS.DLL and OPENAL32.DLL

Hello,
those files are not really a problem as they are somewhat special. The problem is that the DirectX setup didn't reinstalled the file "d3dx10_43.dll"

Ok I also did what Adam said and it told me this. Windows Resource Protection did not find any integrity violations. Then I reinstalled d3dx10_43.dl and i got this error instead 0xC000007B - STATUS_INVALID_IMAGE_FORMAT

Probably the best thing to do at this moment is to restore the file manually as DirectX setup being funny as always :(...

1. In folder where you extracted the DirectX setup that your run earlier should be file "Mar2008_d3dx10_37_x86.cab" (note that's x86 not x64!)
2. Open it and copy the file "d3dx10_43.dll" from it to "C:\Windows\SysWOW64".

(Please note that the Dependency Walker reports a different directory than SysWOW64, but that's kind of illusion done by OS.)

Hmm, I found the right folder and the closest thing I see is d3dx10_37.dll

You have downloaded a file from https://www.microsoft.com/en-us/download/details.aspx?id=8109 and extracted it to some folder. It contains the files of DirectX setup.

Yes, I found it but I only see d3dx10_37.dll

Is there maybe a site that would allow me to show you what on my screen to make it easier for both of us?

I'm sorry, the archive is Jun2010_d3dx10_43_x86.cab and it contains d3dx10_43.dll.
Please accept my apology, it's close to midnight and I'm just an old programmer O:)

Wow it works great thanks so much for your help you've been extremely useful I can finally play now. Thanks again.

Glad to hear that it works, enjoy the game. Sorry it was that complicated.