Sprayer. There shouldn't be a foggy look because the wall and ceiling paint will be backrolled. So, in the end it will look as if it were brushed and rolled. The sprayer is just a tool used to get the paint on faster. If you are worried about overspray getting into the furnace and water heater room, then you should plastic that area off or let the painter know that you do not want any overspray getting in there and he will do it.

Thanks, Paintguy. He will not be backrolling - says he does such a good job, there is no need. We are worried about this.
Also, should we have full prime on walls or spot prime? They were painted 20 years ago but don't look bad except for some stains and holes that need patching.
Will 2 coats of paint using a sprayer look as "good" as two using a roller?

I have never tried to not backroll wall paint in my life, so I can't really be sure how it would look, but my first guess is that it will look flashy. I don't really spot prime anything. It is either full prime or no prime. Since paints are self priming these days, using the actual paint to spot prime is better than using an actual primer. If you use a real primer to spot paint areas, then you run the risk of those spot primed areas flashing through. Usually, two coats of a high quality paint is good for walls. A primer plus two coats is the best though, but in my experience most normal people do not get the best bang for their buck by paying extra for that primer coat.

Really appreciate you taking the time to give such a thoughtful and helpful reply,Paintguy.

You said "A primer plus two coats is the best though, but in my experience most normal people do not get the best bang for their buck by paying extra for that primer coat." Do you mean two coats of paint alone will do ( self-priming paint)? Sorry for my denseness!
Thanks again!

Yep, two coats of paint is usually fine.

Also you should paint with appropriate paint, masonry with masonry paint, metal with metal paint and also wooden bits with another type of paint.. Its not just to spray all over with some emulsion :)

Here is a link that might be useful: Painters and Decorators North London

Thanks, Paintguy. He will not be backrolling - says he does such a good job, there is no need. We are worried about this.

As well you should be. I have never seen a "good" job without backrolling.

Thank you very much for your replies, Paintguy, Rimce44 and Christophern. We decided not to go with this painter and are looking for someone else who will do it 'properly' :-).