It's all personal preference but our flights usually leave out of Buffalo very early, so we travel down the day before and stay on a park and fly package, where the hotel shuttle takes us to the airport and we leave the car at the hotel. ; Sleep Inn is good, or Comfort Inn, several good choices at/near the airport.
We have parked at the airport before, check the Buffalo airport website for a coupon to reduce the cost.
We live in central Ontario, and use the 407 to the QEW to Fort Erie and cross there, or any of the other crossings if one is busier than the others, (Lewiston or Rainbow Bridge - there is an 800 number on the highway signs, or stop at the Casablanca Blvd. rest stop and they have visitors info there re: bridges and Buffalo info)
Not sure on the parking with car seats? ; Are you taking them with you?
As we are at the hotel and just jumping in their warm van in the morning we leave our winter coats and boots in our locked vehicle and just suffer the cold in our light jackets or sweater until we're in the airport. ;
Lots of information on the web, check out the Buffalo airport website also.