First, I want to ask a question about ‘no sooner than’ and ‘no sooner did’. How do we use both sentences? I read a sentence in which both sentence were used but I didn’t get it.
“No sooner had we slept than their singing woke us”.
“No sooner did we catch the sight of the zebra than we shot it”.
The pattern in your sentences is “no sooner did/had X than Y”. This means that Y happened immediately after X.
You can also use “no sooner than” by itself, in which case it means the same as “no earlier than”. For example: “I will arrive no sooner than Wednesday”.
“No sooner had we slept than their singing woke us” sound odd to me. It would normally be “No sooner had we got to sleep / fallen asleep …”.