I recently purchased a house. Underwriting will absolutely ask for your scanned green card. Even though you can function without a valid GC in your everyday life, you would still need to prove your status for a loan. The n400 doesn't constitute as a proof, as some suggested, because it's just an application that can be denied.
I imagine that you can still purchase the house within the remaining validity of your card. It would check the box for them, even if it expires the next day.