Why is "Die Frau spaziert ..." bad in this case?
I don't consider 'läuft' to be a good translation in this case as it could also mean 'to run'. 'Die Frau geht und der Mann rennt' would be better.
'gehen' = 'to walk', 'spazieren' = 'to stroll'
Why zu Fuß is required when geht already means walk?
@davidyahoo: Perhaps it depends on context. It could be that here they try to emphasize that the woman is going on foot, and not going by car for example.