Abaddon's big weaknesses is kiting. He can purge CC effects with Shield and/or Ult, but then you have to go through the cast animation. He's got nothing to handle enemies with higher speed or constant CC (Frost Arrows, Orb of Venom/Skadi).
He looks like he'd be a good trilane support, maybe even an aggressive trilane support in conjunction with a #5 stunner. Shield lets your team be extra-reckless, Death Coil provides good sustain, and one point in the passive gives a reliable 15% speed boost to your team and 5% slow on the enemy. I'm seeing him working a lot like Undying, but trading the awesome pushing power of Tombstone for the superior killing potential of Curse and the extra survivability of Shield + Coil (vs Soul Rip). Get the stunner to disable a target, then rush in and hit them once. If the enemy team tries to focus someone, shield them followed by a coil.
Abadon's debuff works on denies too. I feel that it's a bug. It allows for too much lane control, buffing enemy creeps.
Oh god, that's ridiculous. Also possibly risky (giving the enemy heroes a 15% speed boost is never a good thing!), but still interesting. The wiki claims it also works on towers, meaning the note I made above about "not pushing as well as Undying's Tombstone" might not be true - hit a tower once, and the creeps get 10% more attack speed!