Actually, I doubt I'll be able to help you much. I have no experience with robotics.
I really just replied to try to point out how vague that specification was, because if there IS anyone here with robotics experience, I assume they would expect a much better specification than that.
Now that you ask though, I have a few ideas. As I said, I have no experience in this area, so these are just suggestions.
The specification doesn't say anything about the surface that the robot runs on, does it? In that case, it might be a good idea to use toothed wheels running on a surface that consists of wide teeth that mesh with the wheels. This will ensure that your robot always faces straight forwards. If you make the teeth square instead of triangular, you should be able to get a lot of acceleration and deceleration without any slipping.
You would need a reasonably long robot - I'm thinking around 6~8 cm from front axle to back axle, for stability. Presumably it would have four wheels, and a chassis that sits as low as possible - definitely below the wheel axles. The wheels should probably be smallish, say 1 cm diameter or somewhat less. This is all based on gut feelings and may be wrong!
All four wheels will turn identically. I don't know whether two wheel drive will be enough. Perhaps drive on the front axle and braking on the rear axle. But the braking needs to be controlled, just like the acceleration. So perhaps the front and rear axles should be coupled together.
A stepper motor should be good for wheel drive, but it needs to be small and light. Floppy disk drives use a very small stepper motor to seek the head inwards and outwards.
You would probably want a small microcontroller to control the movement, though it might be possible to do it with discrete components. I should be able to help with the design here.
As for batteries, you can probably get a few minutes' run time from a stack of 3V lithium button cells, which are small and light and easy to stack. Probably an operating voltage of 6V would be good, and you could parallel up several stacks to get the required current.
You need to figure out how to control the robot. It needs to be able to detect the presence of a doorway, so it knows when to stop, and it needs to be able to tell whether the door is open or not. Probably some very light whisker-style wires would do this. You would want at least one on the front, to detect the presence of the door, and probably one on each side, to detect the doorway. They would have to be light enough that they would just bend and spring back as the robot goes through the doorway.
Other options would be ultrasonic sound or infra-red light, but these would complicate everything, and shouldn't be needed. You could perhaps use magnets on the floor to signal the robot that it's approaching a doorway, but attaching it to the doorway so the examiner can move it around might be impossible.
As for the doors, I would definitely go for a sliding door rather than a hinged door, because they're probably easier to actuate, and the robot won't have to stand clear of them. But does it slide left/right or up/down? At least up/down means you only need to move it in one direction; gravity will do the other direction.
I would be happy to discuss ideas with you if you want, but remember that I don't have any experience with the mechanical stuff, and you'll probably need to do a lot of work yourself.
