So, the question is... When I prepare a 3rd party paper for profiling is it just a question of pot luck as to the media setting
Jamie...
I think you have the right idea.
I used to profile my own using Profile Prism, and then look for bargain papers from Office Depot and even at Computer shows where the package of paper was cellophane, no name.
The game was on!
The plan was to try different settings, and write down what I was doing each time.
I tried for the No Color Management setting in the driver, using Epson Prem Glossy. I found that used the least ink. Usually came out the cleanest.
Nevertheless, try Luster, Glossy, semi matte.... mark the target prints, and let your "guys" decide which one will produce the most accurate profile.
As for the other question, I couldn't swear for it, but in my heart, I believe that Canon biases their papers and their printers to match, and try to make a Canon printer use Canon paper.
I don't know about HP.
Best I can offer. All guesswork, I'm afraid.
Fred