Being countered doesn't appear to be requirement for the creatures. I believe it use Last Known Information and give him the creatures anyway.
This is correct.
Most commonly, when people talk about an ability fizzling they are refering to "Countered upon resolution, because all targets no longer exsist" Now dovescape doesn't target, therefor it won't 'fizzle' in that sense.
Also as you elluded to, it only has 1 complete trigger; and the event that causes the trigger is simply 'playing a non-creature spell.' So the entire trigger resolves as much as it can. Also if you play somehting that is uncounterable like Urza's Rage, you would get Birds in addition to the spell resolving.
As a side note: --IF-- dovescape consisted of 2 triggers, then it wouldn't work {but remember this is not true}
So sometihng like:
When a non-creature spell is played, counter it.
When dovescape coutners a spell, make some birds.