At the end of the auction the existing tenants are offered to renew at the winning price first (maybe with some discount to cover the cost of switching tenants). If they don't want to pay that, then it is offered to the auction winner.

I don't think it would work if the auction itself binds you into the rental contract, as you probably want to bid on multiple properties so you have another option if you don't win.

Does anyone know in general why rental properties aren't negotiated further in advance? When I was renting this was actually the biggest issue - when you are struggling to find a place and you have 2 weeks to do it.

Because it’s a game of chicken between the landlord and the tenant. The landlord usually has some rules for notification (e.g., 30-60 days before expiration) and the tenant usually has something similar for notice to leave.

Nothing stops either from notifying or agreeing to a longer agreement before the old one expires. Though with larger buildings (vs individual owners) it’s less likely they’d be willing to diverge from their standard operating procedures and timelines.

Define "in general". All big landlords (Greystar and such) want you to sign lease extension 3-4 months before the current term ends where such thing is legal. There are states that don't allow long term leases past the first term though. For example, in California all leases go month-to-month after the first lease term but they put required notice periods on rent increases (30 days and 90 days if more than 10%).

The exception is if you are in a college town, when many apartments/houses are pre-leased starting before spring break for the fall. I see that all the time in my current college town.