Abu Dawud

Benefits and Issues:
➊ Praising Allah in the manner described above upon sneezing is an emphatic Sunnah.
➋ Giving supplications (duas) to one another is also an extremely emphasized Sunnah.
➌ And the person who does not himself say "alhamdulillah," should not expect a supplication in response from his brother.
➍ As will be mentioned in the next hadith, 5039.
➎ Similarly, it is not necessary to repeatedly respond to a person suffering from a cold or similar illness.

Hadith Commentary:
May Allah keep you on the straight path and rectify your affairs.

Hadith Commentary:
(1)
The first etiquette for the one who sneezes is that he should keep his voice low, as it is mentioned in the hadith that when the Messenger of Allah (sallallahu alayhi wa sallam) would sneeze, he would place his hand or a cloth over his mouth and would lower his voice.
(Sunan Abi Dawud, Al-Adab, Hadith: 5029)
(2)
Some people, upon sneezing, deliberately exert themselves and try to make a commotion, which is contrary to etiquette and is an un-Sunnah act.
The second etiquette is that he should say "Alhamdulillah" (all praise is due to Allah).
The third etiquette is that he should supplicate for the one who responds to him with the words "(Yahdikumullah wa yuslihu balakum)"—that is, "May Allah guide you and rectify your affairs." However, the person who himself does not say "Alhamdulillah" should not expect a supplication from his brother.
(3)
In one hadith, the words "(Yaghfirullahu lana wa lakum)"—"May Allah forgive us and you"—are also narrated as a response.
(Sunan Abi Dawud, Al-Adab, Hadith: 5031)
But this narration is weak in terms of its chain of transmission.

Takhrij:
[بخاري : 6224],
[بلوغ المرام : 1245],
[تحفة الاشراف : 423/7]

Benefits:

The Response to Saying Alhamdulillah After Sneezing:

➊ Regarding sneezing, the Messenger of Allah sallallahu alayhi wa sallam said: When one of you sneezes, let him say «الحمد لله», and his brother or companion should say to him «يرحمك الله». When he says «يرحمك الله», then he should say «يهديكم الله ويصلح بالكم»: May Allah guide you and rectify your condition. [بخاري عن ابي هريره رضی اللہ عنہ]

➋ Upon sneezing, at the very least, it is obligatory to say "Alhamdulillah" (all praise is due to Allah). If along with this, «على كل حال» is also recited, it is better. In Tirmidhi, it is narrated from Abu Ayyub radi Allahu anhu that the Messenger of Allah sallallahu alayhi wa sallam said: When one of you sneezes, let him say «الحمد لله على كل حال». See [صحيح الترمذي۔ 2202]

➌ Abdullah ibn Mas'ud radi Allahu anhu said: When one of you sneezes, let him say «الحمد لله رب العالمين», and the one responding should say «يرحمك الله», and he himself should say «يغفرالله لي ولكم». [صحيح الاسناد موقوف صحيح الأدب المفرد 715 - 934] This is the statement of Abdullah ibn Mas'ud radi Allahu anhu.

➍ Ali radi Allahu anhu said: Whoever, upon hearing a sneeze, says: «الحمد لله رب العالمين على كل حال ما كان», he will never suffer from toothache or earache. [الادب المفرد للبخاري 949]

Hafiz Ibn Hajar rahimahullah said: Its narrators are trustworthy, and such a statement cannot be said based on opinion, so it is considered as having the ruling of marfu' (attributed to the Prophet). [فتح الباري]
However, Shaykh al-Albani said that it is weak, because it is narrated from Abu Ishaq al-Sabi'i, and he became confused in his old age. Therefore, even Hafiz did not declare it authentic. [ضعيف الادب المفرد 926/148]

➎ Rifaa'ah ibn Rafi' radi Allahu anhu said: I prayed behind the Messenger of Allah sallallahu alayhi wa sallam, and I sneezed, so I said:
«الحمد لله حمدا كثيرا طيبا مباركا فيه مباركا عليه كما يحب ربنا ويرضى»

"All praise is due to Allah, abundant, pure, blessed praise, upon which blessings are sent down, as our Lord loves and is pleased with."
When the Messenger of Allah sallallahu alayhi wa sallam finished the prayer, he said: Who uttered these words in the prayer? He asked three times. I said: I did. He sallallahu alayhi wa sallam said: By the One in Whose hand is my soul, more than thirty angels rushed towards it to see which of them would take it up. [حسن، صحيح ابي داود 700 و صحيح ترمذي 331]

From this it is known that if one says these words upon sneezing, there is even greater reward.

➏ If one sneezes during prayer, he should definitely say «الحمد لله», as mentioned in the above hadith. However, it is not permissible to respond to another person during prayer, because that is addressing another person, and the Messenger of Allah sallallahu alayhi wa sallam said: In prayer, nothing of people's speech is correct; it is only glorification (tasbih), magnification (takbir), and recitation of the Qur'an. [مسلم]

➐ If a person does not say "Alhamdulillah" upon sneezing, one should not say «يرحمك الله» to him. [بخاري باب لا يشمت العاطس اذا لم يحمد الله]

➑ If a non-Muslim says "Alhamdulillah" upon sneezing, one should not say «يرحمك الله» to him. It is narrated from Abu Musa radi Allahu anhu that the Jews used to sneeze in the presence of the Messenger of Allah sallallahu alayhi wa sallam, hoping that he would say «يرحمك الله» to them, but he would only say to them: «يهديكم الله ويصلح بالكم». [صحيح ابي داود 4213 و صحيح الترمذي]

➒ From Abu Hurairah radi Allahu anhu, both his statement and the saying of the Messenger of Allah sallallahu alayhi wa sallam are narrated: Say to your brother «يرحمك الله» three times. If it is more than that, then it is a cold (illness). [صحيح ابي داود۔ 4210]

➓ Abu Hurairah radi Allahu anhu said: When the Messenger of Allah sallallahu alayhi wa sallam sneezed, he would place his hand or a piece of cloth over his mouth and lower his voice when sneezing. [صحيح ابوداود 4207 و صحيح الترمذي]

⓫ The reason for saying "Alhamdulillah" upon sneezing is that sneezing expels the blocked waste and vapors from the brain, and removes obstructions from the veins and muscles of the brain, thus protecting a person from many dangerous diseases.
It is narrated from Abu Hurairah radi Allahu anhu that the Messenger of Allah sallallahu alayhi wa sallam said: Allah loves sneezing and dislikes yawning. When one of you sneezes and says «الحمد لله», it is a right upon every Muslim who hears him to say «يرحمك الله» to him. And yawning is from Shaytan; when one of you yawns, let him suppress it as much as possible, for when he says "Ha," Shaytan laughs at him. [بخاري كتاب الادب]
In Sahih Muslim it is stated: When one of you yawns, let him place his hand over his mouth.

Takhrij:
«أخرجه البخاري، الأدب، باب إذا عطس كيف يشمت، حديث:6224.»©Explanation:
From this hadith, it is understood that when one sneezes, one should say "Alhamdulillah" (All praise is due to Allah), and the listener should respond to it.
And the response should be given up to three times if the person sneezes; if it is more than that, then it should not be given, because the Messenger of Allah sallallahu alayhi wa sallam said: "Say to your brother 'Yarhamukallah' (may Allah have mercy on you) three times.
If it is more than that, then it is a cold (i.e., an illness)."
(Sunan Abi Dawud, Al-Adab, Chapter: How many times should one respond to the sneezer, Hadith: 5034)