Well - it's some years later and.. it still seems to depend on the game the most. I play recently almost exclusively offline and connect gaming machine to Internet once every few weeks. E.g. in 'Botanicula' - with almost all achievements gathered on the way - I have only the second 1/2 of them on Steam (some of the missing ones being impossible to miss). :)
'Bejeweled 3' seems to sync all achievements perfectly (with their progress also!) on next online session. For 'Plants vs. Zombies' I have only 1 ('Home Lawn Security') on Steam while I have around 3/5 of them in game itself.
There are many threads around the web regarding those games AND achievements, but the common conclusion is: Some games just don't sync them properly and that's it. :(
The short answer is that OliverBM's assumption seems to be correct and DLC characters are not counted towards these milestone achievements. I would've preferred more authoritative sources, but posted user experiences was the best I could do for most of them.
According to IGN, the Arcade Champion achievement does not include the DLC characters, but does include the unlockable characters:
Tag Ladder does count: You get this achievement when you beat the standard or tag ladder with each of the non-DLC 27 characters (28 on the PS3 version). You can use as many continues as you wish, and you can do this on any difficulty level.
According to this answer, Finish What You Start! does not include DLC characters, but note from reading elsewhere that it does include the unlockable characters:
And no, DLC characters dont count towards the trophy.
According to this thread on xbox360achievements.org, My Kung Fu Is Stronger does not include the DLC characters, but does include the unlockable characters:
-#112, just to be absolutely clear, even if I have the DLC characters, it only counts towards the originals?
-that is correct
According to this post, e-X-cellent!, The Grappler, and Ultimate Humiliation do not include the DLC characters**, but from reading elsewhere do include the unlockable characters:
It seems that DLC characters are not needed for trophies, I got milestone trophies (Finish What You Start !, Ultimate Humiliation, The Grappler, e-X-cellent...) without playing with Rain, Kenshi, Skarlet and Freddy.
Best Answer
In contrast to what TrueAchievements states, it is necessary to finish the Double or Nothing race and complete the next mission where you Meet the Broker to earn the Double or Nothing achievement. This was not an issue with Xbox Live, but rather that I didn't complete enough missions to close out the chapter and earn the achievement. This also answers my question of whether there is an achievement for finishing Chapter 4: Open Skies. The Double or Nothing achievement meets both criteria.