Well, in modern day, Muslims are like you, and me. As far as I know, they have the same rights as we do (in the United States of America). So to answer your question, they buy the house.
(a fun fact about Muslims in modern day Mecca, is that if you steal rom that place, your hands will be chopped off. And if you attempt murder, you throat will be slit. Sorry if that is too much information... I just know a good amount of Muslims in my local community.)
In houses.
Christmas is a religious holiday celebrated by Christians so most Muslims would not celebrate it or decorate their houses for it.
they decorate their houses in whatever pattern they like
Muslims of course mosques are for Muslims Muslims own moques
They rent and own houses
They lived in and still live in houses as the other people live in.
Sure. There is nothing against having trade relationships with non Muslims except for food purchased from non Muslims that is allowed for Muslims to eat per Islam religion.
Yes. They do. They were often small cottages or long houses
we have houses so we are not on the streets and have something that we can call our own
pioneers built their own houses
Lots of Irish people own big houses in Ireland. Lots own smaller ones.
In houses all over the world, just like everyone else :P