You can't sleep in any of the beds in Beggars Row, they're all owned.
Bruma was very cold as well, there were still beggars sleeping in the streets.
Back in 3rd Era Cyrodiil His Imperial Majesty Emperor Uriel Septim V created a group dedicated purely to helping and feeding the poor, this organization called the Soup Brotherhood provided soup and bread to the homeless in Imperial cities, but after the death of Uriel Septim VII and his heir Martin Septim during the Oblivion crisis, Chancellor Ocato cut funding. It has been harder and harder for the homeless throughout the empire since that moment, since the great war all attention towards the homeless has dissapeared. What money the government still has is spend on defence instead of the people.
Thankfully a lot of people died during the great war, so more houses became available and there were less homeless alive anyway.