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Wouldn't Birch be a lot more $$$?

I know your not going to want to hear this, but I question the idea of using liquid nails and staples to hold this thing together. If you have a drill, you can put some 3/4" screws in that will give you a better structure.

what to fill any imperfections and staple indentations with?
Wood putty.

what saw blade for cutting luan?
Fine blade in a circular saw

proper primer and paint? Need a very low gloss paint.
I would prime with standard Killz. 2 coats. The wood is going to soak up the first coat, so don't be shy with the primer. Let the first coat dry really well, knock down any big nasty rough stuff with sanding block, then another thin coat. Final coat with a flat latex.

Get a Wagner power painter or roller and block sand and buff?
I would roll it. Your not really talking about much surface area. A Wagner power painter is going to cost you more time fiddling around getting it to work. When priming, after the first coat, take the roller and put it in a plastic shopping bag, then roll the bag tightly around it. When you are done priming, pitch that roller and move on to paint.

I'd love to cover each panel with a plastic countertop type material, but can't seem to find what I'm thinking of.
Are you REALLY sure you want to use Luan? Go to a proper lumber shop and see what they carry in plastic coated masonite board. This would be in the 'paneling' section. Lots of options there, including products with a very flat, paint ready coating. You can get Formica at Home Depot/Lowes, but do you really want to go that route? Added weight, and frankly, I think it would look chintzy. A flat color would be better.