I'd paint the ceiling, unless it's like new it will look out of place with fresh walls and floors.
Installing some down lights
I'd do this first because of gravity - dirt falls down - you don't want that on brand new carpets, even the walls might get dirty so do it before the paint. No big deal painting around recessed lighting.
Changing the carpets in the rooms
Paint, then carpet - obvious reasons. It's great if you can paint on carpets that are getting torn out. Carpet is better than dropcloths because it soaks up any paint you get on the soles of your shoes.
Installing New blinds
I'd do this last, in fact I'd wait until the paint and carpets are in to pick out your blinds. It makes minimal mess. You don't want to paint around them.
Always clean up last, but once you paint and recarpet, what are they even going to clean?