Let's say one girl is crying in front of her house door because she can't get into her house. Incidentally, her neighbor sees her crying there and comes to ask what happened. He asks" why are you crying?". She responds, "I lost the key to the front door". He asks again, "Did you lose your key?. In this situation, I am wondering if I could say " I have lost the key" instead of "I lost the key". "Have you lost the key?" instead of " Did you lose the key?" If possible, what's the difference?
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I lost the key - Past Indefinite Tense (We use the past simple for past events or actions which have no connection to the present.)
It suggests that either the key has been found or you have otherwise adapted to the loss.
I have lost the key - Present Perfect Tense(We use the present perfect for actions which started in the past and are still happening now OR for finished actions which have a connection to the present.)
"I have lost the key," suggests that you are still in the "lost the key" state, which remains unresolved.
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The have is optional, but if the girl is crying, she will probably choose the most succinct form. In fact, I believe she will not talk about losing the key, she will say
I can't find my key!