Does IMDb provide an api or a data dump to get all title IDs?

As mentioned, IMDB does not have a web service. imdbapi works by screen scraping. The flat files available for download are a legacy from IMDB's pre-Amazon days, and the information there is incomplete. (You could not build your own IMDB with just the files that are available)

However, Rotten Tomatoes does have a nice web interface that returns, among other things, the imdb id of the films - in the alternate_ids section. So, you could use the Rotten Tomatoes API to obtain the imdb id without screen scraping imdb directly.

Rotten Tomatoes' database is less extensive than IMDB's, but it does a pretty good job with modern (1995+) US releases.

You can try TMDb. You can check the documentation here - http://docs.themoviedb.apiary.io/. You can use append_to_response and can get the IMDB id along with other details. There is an alternate solution that I know of. The solution is in PHP. Here is the link - http://blogs.niteshapte.com/2013-07-01-how-to-fetch-movie-identifier-id-from-imdb-using-php.htm. You might need to tweak it as per your requirement.

You can use IMDB API located at

www.omdbapi.com

You can search for movie names like this -

http://www.omdbapi.com/?t=hangover%202

{
"Title":"The Hangover Part II",
"Year":"2011",
"Rated":"R",
"Released":"26 May 2011",
"Genre":"Comedy",
"Director":"Todd Phillips",
"Writer":"Craig Mazin, Scot Armstrong",
"Actors":"Bradley Cooper, Zach Galifianakis, Ed Helms, Justin Bartha",
"Plot":"Right after the bachelor party in Las Vegas, Phil, Stu, Alan, and Doug jet to Thailand for Stu's wedding. Stu's plan for a subdued pre-wedding brunch, however, goes seriously awry.",
"Poster":"http://ia.media-imdb.com/images/M/MV5BMTM2MTM4MzY2OV5BMl5BanBnXkFtZTcwNjQ3NzI4NA@@._V1_SX320.jpg","
Runtime":"1 hr 42 mins",
"Rating":"7.1",
"Votes":"13547",
"ID":"tt1411697",
"Response":"True"
} 

You can get the "ID" (tt1411697) from this JSON response.