Both are possible, but „chodzić” works better because it implies lack of destination – if the cat in question is just wandering about the bed to test where is the best spot to lay down, it should be „chodzić”.
If, on the other hand, it is walking on the bed to get to the bedside table and from there jump on the curtains, to finally arrive close to birds cage(:P), it would be „iść”.
„Chodzić” can be used for present progressive, but only when you are wandering around aimlessly(ie. there is no destination)… Verb aspect in Slavic languages is really merciless and takes no prisoners like that, I'm afraid. ;)
I wouldn't say uncommon, but rather implies a different type of movement. "Lets walk around the park" usually means lets walk in a fairly uniform manor, most likely a circle (note circles are round- roundabouts are circular roads). "Lets walk about the park" would be more like lets go to the park and start walking in whichever direction we feel like, without thought for the route or period of time we wish to be there.
You would probably say lets walk around that lake, but lets walk about that field (unless you mean around the edge of the field, or wish to become quite wet).
In fact, further to my other comment, 'the cat is walking around the bed' would mean that the cat is walking a route that goes around the bed, but never on it. You could, however, say: 'the cat is walking around on the bed'; but 'on' is absolutely essential or the meaning is completely different.
We wondered about it and decided that although "over" shouldn't actually be accepted here (removed), it's not because it means the cat is floating, but because it's too directional -> "is walking from one side to another", not just "walking around". So that would need to have "idzie", not "chodzi".