which includes the following:

Should the words of tashmit (asking Allah to bestow his mercy on a person who sneezes) be said to a person once or after every time he sneezes?

Lexically, tashmit is supplicating Allah seeking His kindness and blessings. In Islamic law, it is supplicating Allah for one who has sneezed by seeking His mercy for him. It is a sunna for a person who sneezes to praise Allah the Almighty after which his Muslim brother is to pronounce the words of tashmit by saying, "Yahdikum Allah wa yusliha balakum [may Allah grant you guidance and peace of mind]". This is based on the hadith recorded by Bukhari in his Sahih through Abu Huraira (may Allah be pleased with him) who narrated that the Prophet said, "When any of you sneezes, he is to say, 'Al-hamdu-lil-Lah' [praise be to Allah] and his fellow brother or companion is to tell him, "Yarhamuk Allah [may Allah have mercy on you]. Then the person who has sneezed should reply and say, 'Yahdeekum Allah wa yusliha balakum."

If someone sneezes more than twice or thrice, this means he has a cold in which case, we must follow the Prophetic guidance and supplicate Allah for him with a suitable supplication by asking Allah for his recovery. A person must not pronounce tashmit to another who sneezes more than twice or thrice. The legal premise for this is the report recorded by Muslim and al-Tirmidhi through Salama ibn al-Akwa' (may Allah be pleased with him) who said, "The Prophet said the words of tashmit to a man who sneezed twice by saying, 'May Allah bestow his mercy on you.' When the man sneezed a third time, the Prophet said, 'This man has a cold' " [The report is in the words of al-Tirmidhi].
Allah the Almighty knows best.