The correct scholarly view is that doing ghusl after washing the deceased is mustahabb, but it is not obligatory.
This is the view of Ibn ‘Abbaas, Ibn ‘Umar, ‘Aa’ishah, al-Hasan al-Basri, Ibraaheem al-Nakha’i, al-Shaafa’i, Ahmad, Ishaaq, Abu Thawr, Ibn al-Mundhir and others, and it is also the view regarded as most correct by Ibn Qudaamah.
See Sunan al-Tirmidhi, 3/318; al-Mughni, 1/134
Shaykh al-Albaani (may Allaah have mercy on him) said:
It is mustahabb for the one who washes the deceased to do ghusl, because the Prophet (peace and blessings of Allaah be upon him) said: “Whoever washes the dead, let him do ghusl, and whoever carries him, let him do wudoo’.”
Narrated by Abu Dawood, 2/62-63; al-Tirmidhi, 2/132. Some of its isnaads are hasan, and some are saheeh according to the conditions of Muslim. Ibn al-Qayyim quoted eleven isnaads for it in Tahdheeb al-Sunan, then he said: These isnaads indicate that the hadeeth is mahfooz.
I say: it was classed as saheeh by Ibn al-Qattaan, and by Ibn Hazm in al-Muhallaa, 1/250 and 2/23-25, and by al-Haafiz in al-Talkhees, 2/134 (Muneeriyyah edition). And he said, at the very least it is hasan.
The apparent meaning of the hadeeth indicates that it is a must, but we did not suggest that it is a must because of two mawqoof ahaadeeth that are to be taken as marfoo’:
1 – From Ibn ‘Abbaas: “When you wash your dead you do not have to do ghusl afterwards, for your dead are not naajis. It is sufficient for you to wash your hands.”
(Narrated by al-Haakim, 1/386; al-Bayhaqi, 3/398).
It seems to me most likely that this hadeeth is mawqoof, as I have explained in al-Da’eefah, 6304.
2 – Ibn ‘Umar (may Allaah be pleased with him) said: “We used to wash the deceased, and some of us would do ghusl afterwards and others would not.”
Narrated by al-Daaraqutni, 191; al-Khateeb in his Taareekh, 5/424, with a saheeh isnaad as al-Haafiz said. This was also referred to by Imaam Ahmad. Al-Khateeb narrated from him that he urged his son to write down this hadeeth.
Ahkaam al-Janaa’iz, 71. 72.
This is what is regarded as most likely to be correct by the Standing Committee (1/318), and by Shaykh Ibn ‘Uthaymeen in al-Sharh al-Mumti’, 1/295
With regard to washing the clothes, there is no basis for doing so in the Sunnah, and it is neither obligatory nor mustahabb.