Are the two sentences correct?
I've lost my key, but luckily I've found it again.
I've lost my key, but luckily I found it just now.
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Neither is correct.
"I've lost" (I have lost) is the present perfect tense. It is not logically possible for you to simultaneously have lost and found your key, which makes the first example incorrect. Neither it is possible to lose your key in the present and then find it in the past, making the second example wrong.
"Have lost" is a little different from, for example, "have opened". You could say "I have opened and closed the door", which would mean you carried out the two actions of opening, and then closing the door. "I have lost" can only mean that something is presently lost, because losing something is accidental and you cannot pinpoint exactly when you lost something - if you could, then you could go right to it and it wouldn't be lost anymore!
So, the only logical possibility is that both the losing and the finding are in the past, which you could express in different ways, for example:
I lost my key, but luckily I found it again.
I had lost my key, but luckily I found it again.
You could shorten this statement a lot - the important thing to remember is that "I have lost" means it is lost now, which it cannot be if you have already found it. So you might say:
I lost, and then found my key.
I found my lost key!
You have to remember that the present perfect has multiple uses, which are used in different contexts, for example:
I have lost my key = I don't have my key now ( you lost your key at some unspecified time and now you don't have a key)
but it can also be used to express experience: I have lost my key many times but I have always found it again or I have lost my key in the past, but that hasn't happened for a while.
I am afraid you have to accept that only native speakers know when certain expressions are used, that is just the way it is. After all, they are surrounded by that specific language all the time. We, as learners, are not.