Assuming the Combined Ops is updated to the latest 1.62 patch, is there a process for installing and updating the GOG release with the DLCs? Do I need to reinstall the 1.62 patch after installing the DLCs?

Yes, because that patch also contains the DLC patches. So it won't install 1.62 again, it will just skip to the DLC updates and install those.
...well, at least this is how it worked with the retail boxed A2CO.

Well, this is all messed up. Tried installing the DLCs then the 1.62 patch, kept getting a file failed to install message. It completed anyway. Tried running the game, got a "Failed to load file "PMC\addons\air_pmc.pbo" - decryption of headers failed" message, the game launched. The menu shows the icons for all the DLC, but none of the campaigns and their missions show up.

I tried running as admin, now I get "Cannot open file G:\Games\GOG.com\ArmA 2\\dta\bin.pbo"

That doesn't sound too good.

Anyway, I'm not sure if the DLCs were patched or not, but you may try to simply run them as mods. Just add " -mod=PMC;BAF " (if you have both DLC) as command line to your Arma 2 shortcut and see if their missions and stuff show up.

You can also try to reinstall the whole thing, of course, running everything as admin.

A completely fresh install doesn't even work. Base Arma 2 launches, but not OA. It gives the same "Cannot open file G:\Games\GOG.com\ArmA 2\\dta\bin.pbo" message.

I've googled around and saw the trick to edit the registry, but that just gives me

"No entry bin\config.bin/CfgInGameUI/MPTable.shadow'. and "Error compiling pixel shader PSSpecularAlpha:0" Lovely!

Well I didn't come across these errors before, so I can't really tell what to do. I would try uninstalling everything and deleting all relevant registry entries and then trying to reinstall again. If you had a working base OA before this you should be able to get back to that someway.

this is what I get if want to run arma2OA.exe or run DayZ from "Play with Six" launcher

"Operation Arrowhead" campaign from arma2.exe works without problem

It was a complete wipe of Arma 2; uninstalled, removed folders (even in appdata), and removed any remaining registry keys. I ran the GOG Arma 2 installer, and that bin.pbo message comes up.

A caveat: I had retail Combined Ops installed before this, I uninstalled my boxed versions to install the GOG version and that's when it started giving me the error messages.

This post mentions a pixel shader error, not sure if it's what you're looking for as it's about dayz and not the normal DLCs but might be worth a look

http://www.gog.com/en/forum/arma_series/is_the_arma_ii_release_here_contain_all_that_is_needed_to_be_compatible_with_the_day_z_mod

barny73 entry:
For those having issues with the registry fix I found this and it worked perfect. This is for those who don't want to use regedit, it is quick and easy.
http://deadfast.armaholic.eu/download.php?file=downloads/armaRegFixer.exe

Also if you get the pixel shader error afterwords just enter your product key and that will fix the issue and you can play DayZ.