Praise be to Allah.
What appears to be the case – and Allah knows best – is that anything that is born from crossing a dog with some other animal – even if the latter is taahir (pure) – comes under the same rulings as dogs with regard to its being prohibited to keep it, except in the cases for which an exemption is made in sharee‘ah; it is also haraam to sell it, and its saliva is naajis. We previously explained the rulings on dogs in details in fatwas no. 114030, 41090 and 69818.
A number of the fuqaha’ (may Allah have mercy on them) indicated that that which is born from crossing a wolf with a female dog comes under the rulings on dogs in terms of najaasah (impurity). Ash-Sharbeeni al-Khateeb said, after mentioning the najaasah of dogs and pigs: That which is born from both of them is of the same nature of both of them; that which is born from one of them crossed with the other(hypothetically speaking), or with any other animal that is taahir (pure), such as that which is born from a wolf crossed with a female dog: it is najis (impure) because impurity dominates.
End quote from al-Iqtinaa‘ fi Hill Alfaaz Abi Shujaa‘ (1/92); Mughni al-Muhtaaj ila Ma‘rifat Ma‘aani Alfaaz al-Minhaaj (2/1229)
Al-Ghazaali (may Allah have mercy on him) stated that that which is born (hypothetically speaking) from crossing a dog with a pig, or from crossing one of them with a taahir animal, comes under the same rulings as dogs and pigs with regard to its being invalid to sell such an animal.
End quote from al-Waseet fi’l-Madhhab (3/18)
Ibn Qudaamah stated that dogs and pigs, and that which is born from them, or from one of them, is najis (impure) in and of itself, and its leftover food or drink, and everything that comes out of it, is also najis.
End quote from al-Mughni by Ibn Qudaamah (1/35)
Shaykh ‘Abd ar-Rahmaan ibn Qaasim al-Hanbali stated that whatever is born from a dog crossed with another animal, and whatever is born from a pig crossed with another animal, will be najis (impure). Please see his words in detail in Haashiyat ar-Rawd al-Murbi‘ (1/341)
And Allah knows best.