Item 1: Wedge Sums of Pointed Sets
Follows by definition.
Item 2: Intersections via Unions
Indeed, $A\mathbin {\textstyle \coprod _{A\cap B}}B$ is the quotient of $A\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}B$ by the equivalence relation obtained by declaring $\webleft (0,a\webright )\sim \webleft (1,b\webright )$ iff $a=b\in A\cap B$, which is in bijection with $A\cup B$ via the map with $\webleft [\webleft (0,a\webright )\webright ]\mapsto a$ and $\webleft [\webleft (1,b\webright )\webright ]\mapsto b$.
