You may think of join $A*B$ as a sphere in the product $(\mathop{\rm Cone}A)\times (\mathop{\rm Cone}B)$.

Once you made all this precise in topological category, everything you ask will be evident.

You may think of join $A*B$ as a sphere in the product $(\mathop{\rm Cone}A)\times (\mathop{\rm Cone}B)$.

Once you made all this precise in topological category, the associativity will follow from associativity of Cartesian product.