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.
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.