Wow, a lot of questions there. I don't know what "accurate" means to you, but that Remington should shoot into an inch at 100 yards with good ammunition. For varmint hunting, I consider the .223 a 200-250 yard rifle and that might be stretching it a bit. The problem is not accuracy, but energy loss for the small bullet at over that.
Absolute maximum range, with an elevation of 30 degrees, is between 1.5 and 2 miles, depending on the bullet used.
As to scopes, there are dozens of makers and hundreds of models of scopes, ranging in price from $60 to $2000. For varmint hunting, I would go with at least a 6 power, and some folks will opt for 8 or 10 power. (You can't hit what you can't see, and small animals are hard to see at 200+ yards with a low power scope.)
Some common brands that have a full line that you might want to look at are Burris (pricey), Bushnell (more moderate), Leupold (also pricey), Simmons (fairly high), Weaver (moderate) and Tasco (low priced, decent quality). If the rifle is to be used solely for varmint hunting, I would go with a higher power scope and avoid a variable. If it is to do double duty on both small and medium game, the variable power would be a good choice.
I suggest visiting a well stocked gun shop and looking at the scopes on offer and also at mounts. There are many kinds of mounts, but it is pretty hard to beat the plain old Weaver mounting system.
FWIW, I would also inquire about used scopes, where there are often real bargains. The fact is that used rifles sell well, but most buyers will pay little extra for a scope, so many dealers take them off and sell them separately. That is puzzling, since a rifle can be shot out, but no one ever wore out a scope by looking through it, and any mechanical damage is usually apparent.
Jim
The factory open sights should be more than adequate for that distance. 
