These are examples of inversion.
'At no time' is followed by an inversion of the verb and subject. A similar phrase which requires an inversion is 'Under no circumstances'. Whatever comes after these phrases, the next word has to be a verb - either a modal or auxiliary verb. I think it's a relic of the Germanic roots of the language.
'Little did I know' is a set phrase, which only varies slightly. You could say 'Little did I suspect', and you can also change the subject, e.g. 'Little did we know', 'Little did they suspect'. It's best to think of this as a fixed expression, though, because it doesn't work with most verbs.