I don't think there's a standard "sweet spot" for booking times. They always change depending on fleet utilization. Your best bet is to book as soon as your dates are known. Just don't prepay. Then every week or so, recheck rates. When you get within 2 weeks, check daily.

Also, if you have time, try changing your pickup/dropoff times by an hour or so to see if there's a big change.

As for deals at the airport, just ask but don't expect anything. If they're low on inventory, there won't be much they can do. Book the smallest/cheapest size you can reasonably fit into and hope for something better when you get there.