Why does 了 change in 突然父亲嘘(了)一口气? The sentence is 


  (1) 突然父亲嘘了一口气


and


  (2) 突然父亲嘘一口气


I would have understand them if it's not the for the "突然" part.

My take:


(1) "Suddenly father exhaled (once)" (like "ooooh" and then maybe
died).
(2) "Suddenly father started to perform an exhale." (and after he
performed it maybe he died).


P.S. Sentence (2) As far as I see doesn't have niether of those readings: 


Conteractual. "If something happens/ed, then he has/would exhaled.
Suprise. He can exhale?!
Probable. Well, he can exhale.
Future. He might exhale.
Deductive. He exhale. (I can say that based on some reasoning)
Abilitive. He can exhale.


that 突然 eliminate all those readings and I'm lost a bit.
Previous answer for ref:
https://chinese.stackexchange.com/a/18061/13501

If you look at my previous answer, I explained:

For the first and second, both mentioned that 了 is used to denote the completion of an action.

For the fourth one, I mentioned that 了 is used only to soften the tone.

In the example above, 

(1)突然父亲嘘了一口气

(2)突然父亲嘘一口气

The first sentence is denoting the completion of the action "exhale", because indeed you are talking about things in the past and father has done and ended exhaling.

However, most Chinese native speakers use it, not to express this meaning, but the fourth meaning actually, which is, only to soften the tone.

Despite the fact mentioned, please note that you cannot use 了 after any verb if the ACTION is NOT completed.

One more point to notice:
The first and second meaning of 了 are only applicable to action verbs. For position verbs(有，坐，死...), they should follow the adjective rule, which is, when confronting these verbs, th meaning should be "become (verb status)".
Example: 我有了。 or  他死了。