My favorite leftovers for beef roasts like sirloin and eye of round, is to make French Dip Sandwiches. Pile the beef on a quality roll, and dip in Au Jus gravy from the grocer.
The other thing I like to do with leftovers from beef roasts is to make a breakfast "hash" (can't say that's the right name for it) by cooking potatoes and onions, and then adding diced cooked beef from the roasts. It's a hearty meal that we have all of the time.
Across the grain: I'm not a super pro cook and so my simplistic way of thinking about it is to place the roast so it's longest way is sitting in front of you right-to-left and then cut across the roast.