Just got mine installed. I called their 1-888 number yesterday cause it's been awhile since i got their email with no further action (and i was bored at work). They were quite nice on the phone and booked an appointment for me today between 4-8, guys came at 4 and set everything up. They asked a few questions about other things I would update in the future and also age of appliances.

All done in about 30 mins.

Nope. Thermostat to the furnace and back a few times.

Do you get to pick which brand of thermostat to install or the installer picked for you?

Did you get an audit/review of how you can "improve" your home's efficiency or anything like that?

You get to pick. Unless you really prefer one unit over the other, Nest is white, Ecobee is black. One might go with you decor better than the other (ie not stand out as much).