Just make sure if they accept your low offer you are ready to close. It’s one thing to pay asking and take your time but if go way below asking they will lose face if you walk away.
Write hosts on AirBnB. They will have multiple places for rent (super hosts to start) and tell them your situation. They will let you book and then if you should have a problem will let you cancel.
Not odd if you enter from airport to just be given 30 day without a prior visa. I wouldn’t read into it. I think they were just saying here is your visa.
actually it depends on where as well as when. But for sure you don’t want to be filing late. It’s not the end of the world but it can be a major headache. So my trepidation is based on avoiding the whole headache to begin with.