The Mossberg 500 will give you 50 yards, minimum. You'll have to try different slugs to find the one that works best, but start cheap and work up. You may end up getting 80 yards out of it. It all depends.
The Mosin Nagent can be a great deer rifle. The cartridge is about the same as a 308 WIN.
1) You need to find soft point ammo. That steel case FMJ stuff is probably not legal for deer, and will not provided the expansion you need to bring a deer down.
2) The rifle is known as a shoulder-buster. They built the stock to be worn with many layers of winter clothing. Usually that means it does not fit the average hunter well. Get a slip-on rubber butt pad and the problem will go away.
3) The bullets for Mosin loads tend to be on the big side, 180 grains or thereabouts. If you can find 150 grain. it usually works better on deer. A place near me sells the stuff, but they were fresh out:http://www.aimsurplus.com/product.aspx? ... groupid=40
4) Accuracy is going to be a rifle-to-rifle thing. I have a M44 that shoots minute-of-bushel basket and an M91 that works fine inside 100 yards. The M44 has been a huge disappointment, but I know other folks that have them that are very accurate.