Solution 1:

  1. Would someone (a native speaker?) actually use "doomed to fail" instead of "doomed to failure"?

Yes.

If the first effort failed, the following efforts were doomed to fail.
Jack London, Burning Daylight, Chapter VIII

as well as about 247,000 results on Google Books.


  1. How can this be applied to other verbs

Here's an example sentence from American Heritage® Dictionary of the English Language:

"With the benefit of hindsight, the fans felt that they knew all along that the Red Sox were doomed to lose" (Daniel L. Schachter[sic]).

Solution 2:

William Shakespeare was indisputably (I hope) a native speaker of English.

Ghost: I am thy father's spirit,
Doomed for a certain term to walk the night, --Hamlet, Act 1 Scene 5

So to answer your questions

  1. Would someone (a native speaker?) actually use "doomed to fail" instead of "doomed to failure"?

Yes, I just gave you an example. And here's another one:
Doomed to Fail: The Incredibly Loud History of Doom, Sludge, and Post-Metal by American author J.J. Anselmi

  1. How can this be applied to other verbs (even if it doesn't make a lot of sense in the first place), e.g. "to live"/"life". Would it be "doomed to life" or "doomed to live"?

See example 1.