You can use other paper's Lemma as long as you give the proper credit to it.
This is what science is all about.
All your options are valid, and might fit in different situations.
However, even if you give your own proof, you should give credit to the original proof (stating that your proof is different / better/ clearer).
Maybe to be clear, I usually do this:
- If the lemma is secondary, I just refer to the other paper.
- If the Lemma is important, but the details are not, I cite it and refer to the other paper for details.
- If the Lemma is crucial, I state it formally with all the parameters and details. If the statement I use is exactly the same as the lemma in the other paper, I refer to the proof there. If it is slightly different, I acknowledge the other Lemma and state that "my" lemma is a simple variant of it. I add a proof when needed.