Is "would" used correctly in these sentences?
I would not be surprised if you would lose.
I would not give you the weapon if you would use it to harm others.
It is acceptable grammatically to use the word "would" multiple times in a sentence, but the first example you provided might best be revised to invoke what is referred to as the "subjunctive mood" or sometimes "subjunctive mode."
The word "would" occurs after the word "if" in the dependent clause of your sentence, and because there is the element of possibility in the actions of others implied in the sentence, the first example sentence would probably best be revised as indicated below:
Because there is some element of uncertainty in the matter (the "if" in the sentence suggesting the person's losing versus winning), it is not appropriate to use the word "would" in this sense. In other words, the effects of something (presumably a the outcome of a match or a court trial) on the indirect object ("you," in this case) of this sentence are yet undetermined at the time of the utterance.
The second sentence you provided is a better example of "would" in a sentence, however as written, the sentence is a bit clunky and confusing:
The second instance of the word "would" (even in the revision I suggest above) seems to be out of place for the same reason that the first example sentence does: the consideration that the person might use the weapon to harm others is speculative, and the reality of their using it to harm others is yet undetermined.
There are plenty of good examples of when one might use the word "would" multiple times in a sentence, however it is best not to use the subjunctive form of the verb "to be" (which in most cases is "were" or "be") in order to express uncertainty of actions that have yet to occur.
Yes, it is acceptable to use the word "would" multiple times in a sentence, but given your examples, there are reasons against using it such as to avoid ambiguity. The subjunctive mood allows a speaker to concisely "express various states of unreality," particularly in dependent or subordinate clauses of sentences.