"läuft" should be allowed right?
läuft should not only be allowed! It is in fact more natural in this context...
Why isn't accusative used? In this case, he is moving, so "auf" should be accusative, shouldn't it?
Hmm, but think of it as if he was using it as a vehicle, as a means to get somewhere (e.g. er geht im Auto), his position is not moving regarding the foot. That's how I thought about it, but I can really see how it is a confusing sentence :)
"Auf einem Fuß" means dative case. However in "auf den Tisch" no dative is used. Please help me understand.
Auf den Tisch -onto the table,auf dem Tisch -on the table.The first is accusative, the second dative case.
Is "Das Kind geht am einen Fuß" incorrect?
It's incorrect. "Am" means an + dem. "Dem" is an article so it can't be used with the other one (einen).