This is definitely Stealth Symphony
- Boy with cursed mechanical dragon wings: check, the protagonist
- joining a mercenary group with a talking alligator boss: check
- Vampire Girl: check
- Robot dude with robot dogs: Check, no link found :(
- Stopping a slave-trafficking caravan: Check, that's about chapter 6 - 9
Synopsis from MAL (taken from MangaHelpers):
In a world where elves, dwarves, and invisible dragons live in harmony
inside a metropolitan city, Jig sets out to find someone or something
that can unleash the curse that he carries on his back. As he attempts
to survive in the city of Jinbocho, a city filled with fantastic
secrets and organizations, he encounters many powerful individuals who
set out to either destroy him or save him. Will Jig be able to
discover the true secret of his curse, or will it lead him to his own
demise?