Is it permissible for the Muslim to stay in the house of a non-Muslim and pray there?

It is permissible for a Muslim to stay in the house of a
non-Muslim in the sense of buying it or renting it, and he has to purify
it of whatever may be in it of the traces of shirk and sin such as haraam
images, and impure things such as alcohol and the like. 

With regard to staying in a non-Muslim’s house as a matter of
hospitality and friendship between them, that should not be done except in
cases of necessity when there is no alternative, because of the general
meaning of the words of the Prophet (blessings and peace of Allah be upon
him): “Do not be friends with anyone but the believer and do not let anyone
eat your food but one who is pious.” Narrated by al-Tirmidhi (2395); classed
as hasan by al-Albaani in Saheeh al-Tirmidhi. 

And he (blessings and peace of Allah be upon him) said: “A
man will follow the way of his close friend, so let one of you look at who
he takes as a close friend.” Narrated by Abu Dawood (4833); classed as
saheeh by al-Albaani in Saheeh Abi Dawood and elsewhere. 

It says in ‘Awn al-Ma‘bood: 

i.e., he should think long and hard about the one whom he
takes as a close friend. If a person’s religious commitment and character
are pleasing, he should take him as a close friend; if his religious
commitment and character are not pleasing, then he should avoid him because
he will inevitably be influenced by him. End quote. 

With regard to praying in the houses of non-Muslims, there is
nothing wrong with that if the spot where he is going to pray is clean and
there are none of these images or statues that they venerate or worship in
that spot, because of the general meaning of the words of the Prophet
(blessings and peace of Allah be upon him): “The earth has been made for me
a place of prostration and a means of purification, so wherever a man of my
ummah is when the time for prayer comes, let him pray.”

Narrated by al-Bukhaari (323), Muslim (810). 

So the entire earth is a place of prostration and it is
permissible for the Muslim to pray there, except in cases for which is
evidence that they are excluded, such as graveyards, bathrooms and camel
pens. 

See the answer to questions no.
13705 and
140208 

Ibn ‘Abd al-Barr (may Allah have mercy on him) said in
al-Tamheed (5/227): 

Al-Bukhaari narrated that Ibn ‘Abbaas used to pray in curshes
if there were no images present. Ayyoob, ‘Ubayd-Allah and others narrated
from Naafi‘, from Aslam the freed slave of ‘Umar, that when ‘Umar went to
Syria, one of the prominent Christians made some food for him and invited
him to eat, and ‘Umar said: We will not enter your churches or pray in them
because of the images and statues in them. 

Neither ‘Umar nor Ibn ‘Abbaas objected to that except because
of the statues in them. End quote. 

If the place where one wants to pray is free of statues and
the like, and the spot is clean, it is permissible to pray there. 

And Allah knows best.