I have hardwodds thoughout my house except for this one room I will be doing. My question is whether I have to or should lay hardwoods perpendicular to floor joists. It may be easier for me as far as cutting-wise to laythem parallel to joists but don't want problems down the line. The wood is 2 1/4 " x 3/4' hardwood from Lowes.

The house is 10 years old and in the Southeast US.

Also about how long will it take me (New to flooring) to do a 10 by 13 room with a rented power nailer?

Id figure since you are a rookie its going to take two full days once the room is preped. Someone who knows whats going on its a 3 hour job to nail it up and trim it out with shoe. Dont lay it parallel to the floor joist it will eventually squeak more and might even sag.

I have a related question - How about laying it diagonal to the joists? I have hardwood in to adjoining rooms (oak) that is stained different colors. I'd like the new install to be a third color since I'll never match either of the first two rooms. I figure doing it diagonal will break the rooms up. Thoughts?

45 is actually stronger than perpendicular so yes it is fine. Thats why old historic homes pre plywood have one bys on a 45 deg angle as a subfloor.