OK so I was a little niave in the fact that I thought hunting regulations state to state would be similar.
I suppose the general over riding fact is that hunting in the US is cheap, even if you go to another state.
Cost is relative. Some states only charge for the hunting license, such as Alabama, but they charge a great deal more for an out-of-stater. My dad lives in North Florida and hunts in Alabama because he has a place to hunt there on private land with a lot of quality deer, so for him, the extra cost is worth it. Your license buys the opportunity of two deer per day with no tags required as well as hunting whatever else is legal.
Out West, things change a bit. I'll talk specifics for New Mexico. First you have to buy your license. That is the entry fee to apply for the lottery. From there you apply for everything that you want to hunt, paying the tag fee (refundable if you do not draw) and an additional application fee. The state is divided up into numerous Game Management Units. For each species, there are specific hunt dates and numbers of tags available in each GMU where hunting that species is allowed. You can apply for up to 3 hunts per species specifying the GMU and hunt dates for each hunt. For deer and elk there are "fourth choice" applications as well where you don't specify anything more than the species and quadrant of the state and you can get any "leftovers" after the regular draws are done.
The types of hunts available are pretty varied and some have very specific application requirements. There are youth only hunts where the hunter must be younger than 16. There are muzzleloader and archery specific hunts where you may only hunt with the prescribed weapon type. There are mobility impaired hunts for those with disabilities. There are restricted area and veterans hunts where you have to be a member of the US military, a government employee with an access badge for the restricted area out here or a veteran of the Iraq or Afghan wars to apply.
Species that can be hunted in New Mexico are also widely varied and include wild populations of some African species. We have elk and deer (whitetail and mule deer, both desert and Rocky Mountain), pronghorn antelope, Barbary sheep and javelina that you can apply for and potentially draw a tag for every year. In addition, there are once-in-a-lifetime trophy hunts for several species like oryx, ibex, Rocky Mountain bighorn sheep and desert bighorn sheep. You can apply every year for these until you draw. Once you draw, whether you actually are successful in your hunt or not, you can not apply again for the rest of your life. In the case of oryx, there are also broken horn, and "at large" hunts which are not once in a lifetime but aren't going to yield a trophy mount but they will put some excellent meat in the freezer.
So when you get to your hunting area you are free to hunt alone without any form of guide?
Most hunts are public land, on your own. You can increase your chances of drawing a particular species by signing on with an outfitter (guide) ahead of time and including that contract with your tag application. A certain number of tags for each hunt are reserved for guided hunts in order to support their business.
What happens when you wound an animal and cannot find it, do you surrender your tag or shoot another?
Unfortunately, you may shoot another.
In terms of "deer" available to shoot. You would have the Whitetail and Coue's Deer and the Mule and Columbian Black Tail Deer. Are the two subspecies common to shoot or are they difficult to find?
It varies widely depending on area of the country. In Alabama, for example, the state is overrun with whitetails to the point that you can shoot two a day. In the GMU here in New Mexico where I shot my mule deer this year, they issued a total of 2400 tags for the two rifle seasons this year. Of all my friends that drew a tag, I was the only one that even saw a deer. Success rates for that particular GMU for deer is the lowest in the state and there has been a serious push from the locals here to get the state to limit, or even eliminate, deer tags for a few years and increase the cougar harvests to help the deer population recover. I lived in Montana for a few years and large mule deer were plentiful in some parts of the state and large whitetail in others.
Then you would get the Pronghorn, Elk, Moose.
Which of the species is best tasting?
Which is the hardest to hunt?
Taste? Elk. The buffalo hunters in the 1800's hunted the plains elks species to extinction because they tasted so much better than anything else available. Elk is probably the best tasting meat I have ever eaten, but I've yet to try oryx.
Hardest to hunt? Have to be some of the high mountain exotic/rare species of sheep or goat. Of those you listed, though, trophy elk and moose can be very physically taxing, depending on the region. There are several regions here in New Mexico and one I know of in particular in Montana where there is no vehicular access for dozens of miles in any direction to the hunt areas for elk. Because of the remoteness of the region, the hunter density is low, the trophy elk density is high and the hunts are very demanding of the hunter. You have to either walk or travel by horse or pack mule into the hunting area, hunt and then pack out whatever you kill. The terrain is very steep and the forestation can be very thick as well. The weather may or may not cooperate. In that wilderness area of Montana, grizzly bears are drawn to the sound of rifle shots and have mauled several elk hunters who were dressing out their trophies in the last few years.
Now that is all public land hunting in the West. The private land rules are a bit different here in New Mexico. If you have land owner permission, you can forgo the tag lottery and simply buy a tag for deer, elk or pronghorn "over the counter" for a 5-day period and go hunt. Many outfitters will lease hunting rights on cattle ranches and charge clients to use that access. The fee usually gets you a guide who knows the land and the habits of the animals that live there. They can charge pretty much whatever someone is willing to pay and some charge additional fees for trophy grade animals. The law here is that you can only get one tag for any species in a given year, so if you drew a public land hunt, you can't buy a private land tag and vice versa.
There is one other special hunt in the state, and that is the Valles Caldera Wildlife Management area. Valles Caldera is an extinct volcano crater just west of Santa Fe and is one of the primary breeding grounds for Rocky Mountain elk. The area is a public land trust and is managed by a board of trustees. To raise funds, every year they essentially raffle off all the elk hunting tags in the WMA. Each person can buy up to 20 tickets for the hunts and divide their tickets between the various hunts as they see fit. The specific hunts are drawn such that each hunter is assigned a parcel of several thousand acres of the WMA that they have either sole hunting rights on or share with 1-2 other hunters for the 5-day hunt. Many huge trophy elk are taken out of there every year and the success rate is nearly 100%. It is a remote region and it is a physically demanding hunt, but because of the odds and the very high chance of shooting a massive trophy, I'll probably buy in every year until I'm just too old to hunt. Figure I've got at least 25 more years in me.