You're welcome!

Kate's alternative is good, but no better than your original phrase - personally I would say it your way rather than hers. However, I would say it the other way round: the cafeteria is within easy reach of my dorm. This is because the dorm is your 'base', and you can easily reach the cafeteria from it, rather than vice versa. This asymmetry or directionality means this from of words isn't an exact replacement for 'being near'.

You could, but it would be more likely that you would say the dorm is "a stone's throw" from the cafeteria. "A stone's throw" means not very far away. I keep a glass of water on my bedside table so it is 'within easy reach' during the night. Literally, it takes very little effort to reach out to the glass. The glass would be within reach (but not 'easy reach') if I placed it on the desk across the room from the bed because I could still get it, but I would have to walk to the desk first. It can mean something which is not physical. Healthy food is within easy reach (because fruit and vegetables are cheap and you can buy them anywhere): it is not physically close, but it is easily achieved.