Is it permissible for an imam who found a watch in the mosque to sell it and donate its value, or to wear it and benefit from it, given that more than a month has passed since it was lost and its owner has not been found?

You should not have announced the found item (Luqatah) in the mosque. The person who found the watch is obliged to announce it for a full year in markets, at mosque entrances, and in gatherings of people. If its owner is found, it should be handed over to them. Otherwise, the finder gains ownership after the completion of the year. It is not permissible to dispose of it or benefit from it before announcing it and the completion of the year.

If the finder gives it to a trustworthy person to announce it, their responsibility is discharged. The announcement should be made in places where its owner is likely to look for it, and at the mosque entrance, once every two or three days, either by the finder themselves or by someone they trust.

Therefore, you must announce the watch for a year, as long as its finder gave it to you. It is not permissible for you to benefit from it, sell it, or donate its price. If the year passes and its owner is not found: - If the finder gave it to you because they no longer wanted it, then it is yours. - If they gave it to you to announce it on their behalf, then it belongs to them and must be returned to them.

If you gain ownership of it, then everything you mentioned in your question is permissible for you, including selling it and benefiting from its price.