RE: Advice on rifle
A Stevens M200 in .270 topped with a Nikon ProStaff 3-9x40 scope should come in right around $500..or just a bit more. If you make the scope a Pentax Gameseeker or a Sightron S1, you will make your budget cleanly.
Either will make a nice, inexpensive deer rig that will last for many, many years.