1. Jim promised that (he would never/never would he) tell anyone else.
2.Not until it was too late (I remembered/did I remember) to call Susan
3.Hardly had we settled down in our seats (than/when) the lights went out.
4.Only after checking three times ( I was/was I ) certain of the answer.
5.At no time (I was aware/was I aware) of anything out of the ordinary.
6.Only Catherine and Sally (passed/did they pass) the final examination.
7.Only when (Pete has arrived/has Pete arrived) can we begin the programme.
8. No sooner had it stopped raining (than/when) the sun came out.

In my opinion - 1 - he would never there is no inversion because the sentence doesn't start with adverb)
2- did I remember - because the sentence start with adverb - Not until.
3. when - it is impossible to be than there is no sense
4.I was - my intuition but I am not sure why !
5.Was I aware - I think At no time - requires inversion
6.passed - from the translation of the sentence i understand why it is passed
7.pate has arrived - This choice is based on my inuition again - Please, somebody explains to me when we use inversion and when we don't use .
8.when - from the transation of the sentence
#1-- right
#2-- right
#3-- right
#5-- right
#6-- right
#7-- right, but 'Pete', not 'pate', which is the top of your head.
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That's right I don't know very well the difference between than and when ! Emotion: sad
There is a rule:
No sonner + than
Hardly + when