I may get hung out here by a couple of parties, but I'm okay with that.
Many people on this site believe that it takes a higher end custom, or semi-custom gun to obtain the kind of accuracy needed for long range hunting/shooting. That is not necessarily the case, but as I said, some believe it.
I, for one, have found in my years of shooting that spending money on optics is much more important than spending money on the actual gun. I have three sub $500 dollar guns that all shoot 1/2 MOA or better. They all have $400 or more scopes on them though. In rifle and optics alone you can spend a good deal of money, so the sport is not cheap, even if you take the cheap route.
I appreciate SidecarFlip's posts, and I get his sense of humor, though there are several on this site who may not. He is a wealth of knowledge, and he is willing to help people. He has helped me in the past, so I wouldn't read too much into his post.
It is hard to find one rifle that will do everything, but for the list of things that you have given us I would agree with Mudrunner on the 7 RM. The 7mm's have a good selection of high BC bullets to chose from, and the 7 RM takes less powder than a lot of the larger magnums. Brass can be found, and Ammo is usually available at stores should you forget to bring your own on a trip.
Making the decision to get into reloading will help you to squeeze accuracy from your rifle. I have a 700 SPS in 7 RM that shoots one hole groups at 100 yards, and the best group that I have gotten out of it at 1000 yards is 6 inches. I have tried many different loads through the gun, some shot less than 1/2 MOA, some (mostly manufactured ammunition) have shot as large of groups as 3 MOA. Once you decide on a gun it just takes a lot of work to get it to shoot the way you want it to, and patience to find the loads that it likes.
You can get into a long range rifle with a lower end factory rifle, but you need to be prepared to do your part. From what I have gathered, it is much easier if you chose to go the custom route, though it will cost more. I'm like you, I don't have much money. I had to save for 2 years to buy the scope that is on my 7 RM.
Best of luck. I hope this helps, and I hope it helps to put a couple of fires out too!
Joe