Yep, it can be very confusing because each state does it differently and some of those states are more than happy to take your NR $ but less than helpful with explaining their regs and such. Another factor that turns away a lot of NRs is how states like WY require you give them the full tag cost up front, which could be around $2000 for moose as an example, then hold that $ for several months before refunding it. Then a few other states like UT and ID require you to buy a NR hunting license to even participate in their draw, and more than likely you won't even get to use that NR license, and it's not cheap!
Regarding WY, I think you are better off NOT trying for a type 9 tag. Like Whip said, for most units you are basically applying for an elk rifle tag and most of those seasons start on October 1, but for a few $ you can get an archery tag and the bow hunt the entire month of September. Most WY residents treat the September archery season as a scouting month to prep for the gun season, so the bow pressure is less than in CO, for example, especially during the week. They will set up their camps on the 9/1 opener, leave it up thru October and maybe hit it on the weekends. I know a few guys who drew the type 9 tags expecting a wonderful bowhunt and were very surprised to find themselves sharing the woods with some early-season rifle hunters who drew gun special tags for their unit. Not good!
Another WY anomaly that I think is unique: there is the regular draw and then there is the "special" draw, where for another $400 or $500, I can't remember which, you can significantly increase your odds. Another trick most people don't realize is that there is an early gun season in most units that starts on 10/1 but there is also a late gun season that starts in November - getting that tag is even easier (i.e. Higher draw odds) and for a bowhunter, who cares when the gun season starts because you will be bowhunting in September anyway. Most gun hunters want the early season so that they can hunt rutting bulls so they apply for the early gun season tags.
Lots of little tricks like that can lead to a great tag, once you learn the system.