An American term for somewhere to lay your head for the night (or longer). "Do you mind if I crash on your sofa as I've missed the last train out to the suburbs ?" as an example.
As to finding a willing recipient here to provide a crashpad in one of 3 diverse locations, probably hoping for too much. If on a budget, I'd try local Craigslist sites for each of the desired cities, or one of those 'sofa surfing' sites/groups. Good luck.