If I have a planar graph G and another graph H that's been created by crossing two edges of graph – and its very obviously non-planar. Can I use it as an argument to show that they can not be isomorphic?
Is every isomorphism of a planar graph also a planar graph?
Planarity is a property which is based on the existence of an embedding in $\mathbb{R}^2$, which does not exhibit any edge crossings. For any graph you will find many embeddings, which do not satisfy this property, so disproving that two graphs are isomorphic based on two drawings does not work. And yes, planarity is preserved by isomorphisms.