Is it permissible to say "Yarhamukallah" (may Allah have mercy on you) to a non-Muslim when they sneeze, and is it permissible for a Muslim to say "Yarhamukallah" to himself?

First: The Sunnah is to say to the sneezer, "Yarhamuk Allah (May Allah have mercy on you)," if he praises Allah. If he does not praise Allah, then one should not say it to him.

Second: If a non-Muslim sneezes and praises Allah, one should say to him, "Yahdīkum Allah wa yuṣliḥ bālakum (May Allah guide you and set your affairs in order)," and not "Yarhamuk Allah," because mercy is specific to believers.

Third: If a Muslim sneezes and praises Allah, it is not legislated for him to say "Yarhamuk Allah" to himself, whether he is alone or with a group. Rather, he should only praise Allah. This has not been reported in the Sunnah or from the Companions.