Generalization of results obtained from a paper that may be "false"
What should I do [when a published result contains a mistake]? should I try to find another proof for the theorem and write it in a paper?


You cannot use their result as-is, because it is incorrect. If their result is sufficiently interesting, then you could perhaps publish a paper that corrects their result, otherwise, you could include a correction in your paper before using the corrected result.


  In the paper B, I will use theorems similar to those of Article A, but by weakening the assumptions assumed in A. Do you think there is any chance that this can be considered plagiarism?


No. 

You should make it clear that the theorems introduced in Article A are incorrect, and you should clearly motivate the introduction of your similar theorems that correct the originals. 

This isn't plagiarism. 



The OP clarified their plagiarism concerns as follows:


  my proofs will be similar to those used in paper A, except that I weakened the hypotheses. 


Before stating your proof, you could explain "The following proof builds upon a proof by Original Authors [Theorem X, Paper A]," you could go further and explain "novel aspects appear in the first, third and fourth paragraphs" or "novel aspects will be highlighted in the proof" (with suitable highlighting in the proof, e.g., "this aspect is new," possibly parenthesised).