I've followed these instructions and gotten DH working on my Mountain Lion Mac using my copy purchased from Amazon. I've noticed a couple of odd things.
1 - At no point in the installation process did DH ask me for my key. I have one, but it was never entered or asked for.
2 - I can only run the game when I'm pointing Wineskin at darkest hour.exe, not at darkest hour launcher.exe. So I never get the opening screen that lets me choose mods, shows the version I running, etc
3 - My checksum reads QBGX. DH 1.03 full.
4 - The game runs pretty well. Every now and then I get an error message pop up. Half the time when I click okay on the error is closes DH, the other half it doesn't. I've also had a corrupted save game file, which is the first time that has ever happened to me. I was only 2 years or so into the game, so not a lot was happening, this may not mean anything but it's the first time it's ever happened to me.
5 - Playing as the US seems different. I haven't played much 1.03 but things seem harder (which is nice). For example, not only does Japan conquer China almost without fail (not unusual) but Germany just runs through Russia like the Soviets aren't even there, even when I give them plenty of blueprints & Lend Lease. Bitter peace within 8 months to a year of declaring war.
6 - For some reason after Japan declares war on me, and the UK I cannot join the allies. 0% chance. I've never seen that before. Not sure if that's a new 1.03 thing or if something is up with my checksum.
Is it possible that since I never had to enter my key the game thinks I have a pirated version? To download from Amazon I had to actually download the main .exe file from a work PC since my Mac would download the first .exe file (that then allows you to download the game) and then not be able to do anything with it. Again, I do have a key.