I would also consider staying the night in Lake Louise between days 5+6 instead of going to Banff. Why? If it's a nice day in summer, the parking lots at Lake Louise and Moraine Lake are full before 8 am as mentioned, so you will want to hit those as early as possible on day 6. I would spend the rest of the entire day driving to and from the Columbia Icefields. There will still be a decent amount of winter snowpack in the mountains in June, and again, if the weather is co-operative, it is one of the most beautiful drives you will ever undertake and you will want to stop for a lot of photos on that road.

Sulphur Mountain is an easy 2 hours out of your day in Banff.