Have you emptied your wastebasket?  maybe old deleted version of steam_api.dll is still there.  Perhaps the version you deleted has been modified by using steam, and the 'new; version you downloaded to replace it is not 'modififed' via use... hence steam as it exists on your PC doesn't 'like' the original version of steam_api.dll?  Also, there must be a way to re-install, or 'verify the integrity' of the steam install? 

Reinstall game:  in steam = select game, right click on game, select bottom choice called "properties."  Choose "local files" tab.  Then select "verify integrity of game cache."  This will fix the game.  However, is steam_api.dl a dll for the game, or for running the steam client?  If its for the steam client, then you will have to reinstall all of steam. I have never done this and I'm clueless about how to do it.  May I suggest, before you do reinstall steam, that you have your steam ID, password, and all the codes fop each of your games written down somewhere else, so you can get all your games back after the reinstall of steam.

I'm running win7, and win defender has never tagged any steam dlls - so maybe its win8 still needing to shake out some quirks?  If all else fails - try (errr sorrrryyyy) steam customer support.  (I think it has  a separate log in and a separate password, etc.)  Sorry I cant be more helpful.  

i re installed the game 

and its working properly now  

and started from the checkpoint where i was ...