Borderlands – get each collection achievement from the Claptrap DLC in a single playthrough

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.

I believe most, if not all, of the accumulative achievements are tracked across multiple playthroughs. Like the moose finding achievement, I know I have been using several playthroughs trying to get that and so far it has tracked my progress accross all of them.