Here's one for you:

Although it is not drawn planar, it is planar, and so it has no $K_5$-minor. However it has lots of $K_4$-minors. For example, the $0,4,6,8$ induces $K_4\backslash e$ and so adding any path from $4$ to $6$, say $4-7-1-6$ gives a graph that can be contracted to $K_4$. So this shows that deleting $2$ or $3$ or $5$ still leaves a $K_4$-minor. The remaining cases are similar and left as an exercise!

As for the colouring, it is easy to see that it is not $3$-colourable just by working with the triangles (start with $0$ red, $6$ green and $8$ blue and follow your nose). But deleting any edge leaves a $3$-chromatic graph.

You can see this easily enough by hand, or just run this code:

g = Graph("H?`vAqz")
for e in g.edges():
    h = copy(g)
    h.delete_edge(e)
    print h.chromatic_number()

Here's one for you:

Although it is not drawn planar, it is planar, and so it has no $K_5$-minor. However it has lots of $K_4$-minors. For example, the $0,4,6,8$ induces $K_4\backslash e$ and so adding any path from $4$ to $6$, say $4-7-1-6$ gives a graph that can be contracted to $K_4$. So this shows that deleting $2$ or $3$ or $5$ still leaves a $K_4$-minor. The remaining cases are similar and left as an exercise!

As for the colouring, it is easy to see that it is not $3$-colourable just by working with the triangles (start with $0$ red, $6$ green and $8$ blue and follow your nose). But contracting any edge leaves a $3$-chromatic graph.

You can see this easily enough by hand, or just run this code:

g = Graph("H?`vAqz")
for e in g.edges():
    h = copy(g)
    h.merge_vertices([e[0],e[1]])
    print h.chromatic_number()