Missing binkw32.dll and then missing nxcooking.dll

@Gumniana

Can you right click on the game tile in Origin and then Repair?

Let us know if this fixes those DLL files, thanks.

For ME1 I use Steam, only my ME3 game is on Origin

@Gumniana

In that case, reinstalling the game may be the way to go.

Let us know how you get on with that, thanks.

Thanks but as I said, I did it before and the DLL are still missing

@Gumniana

I do not mean download some DLL from the 'net as that is a likely way to infect your machine with malware and/or viruses. I suggest you update your antivirus and do a full system scan before reinstalling the actual game.

I have an active antiVirus but thanks, I will check that again

But I'm pretty sure nothing will change, as I tried before with other missing DLL : delete everyting and take 3 hours to reinstall the game for having the same error when the game launchs

@Gumniana

Your Anti-Virus software will not help you at all, or at least not until it is to late if you download a DLL from the wrong place. 

Steam game repair: (right click at ME > Properties > Local Files > Verify integrity of game cache.)

Do you use mods on ME1?

If the DLLs are present in the folder and the error still persist, it is most probably a Registry mismatch.

• Close Steam completely.
• Copy the "Mass Effect" folder under "<drive>:\Program Files (x86)\Steam\steamapps\common" to another location.
• Uninstall the game using Revo uninstaller (the freeware version). It will give you options to clean the registry as well, use them, the default setting (Moderate) should be sufficient.
• reboot
• Close Steam if it is set to auto start.
• Copy the "Mass Effect" folder back to "<drive>:\Program Files (x86)\Steam\steamapps\common"
• Open Steam.
• Steam should recognize the folder and should install the game > test.