Is it permissible to give charity with the intention of facilitating marriage, and is it better not to place the charity in a mosque where the integrity of its supervisor is doubted, or to place it?

The default assumption for Muslims is trustworthiness, so it is not permissible to accuse them or harbor ill suspicions about them without clear proof. There is no harm in placing charity in the mosque, but it is preferable to distribute it personally to ensure it reaches those who are deserving. If you have doubts, you can distribute it yourself or appoint someone you trust other than the man in question, but it is not permissible to accuse him of treachery without evidence.