Is it permissible to return the greeting of Salaam and to say "Yarhamuk Allah" (may Allah have mercy on you) to one who sneezes silently, by only moving the lips, given that both are obligatory?

Returning the salutation is an obligation by consensus, and it is an individual obligation (fard ayn) upon one person, and a collective obligation (fard kifayah) upon a group. The reply must be audible to the Muslim, otherwise, it is not considered a reply.

Saying "Yarhamuk Allah" (May Allah have mercy on you) to one who sneezes is a collective obligation (fard kifayah) if they praise Allah. The obligation is fulfilled by one person saying it. The saying must be audible to the sneezer.