I found I could say something like "When did you last go to the cinema?" and "When was the last time you went to the cinema?", but could I use last time in the end of past simple sentence?
Learn English – Is it correct to say “When did you go to the cinema last time?”
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It's legal to say something like, "When did you go to the cinema last?" although I prefer it as in your first example, "When did you last go to the cinema?" Adding "time" to the end of that is not natural English, though.
There are, of course, many ways to end a sentence with "last time", as long as it makes sense in context and (I think) is part of a phrase:
And so on. The difference between these and your question is that "last time" is the direct object of the question, "when was the last time ..." and the rest is an adjectival phrase that modifies "time". With the other examples, "last time" is part of a phrase that modifies some other part of the question.