Pion+ and proton make kaon+ and another strange particle, X. Why is this the strong interaction? 
The answer is the strong nuclear force. 
After having looked at Feynman diagrams online for similar interactions I figured out an interaction for this one that I have drawn above. Is that correct? I believe I understand the interaction that I have drawn but I don't understand why it's impossible to be any other type of interaction.
As K+ has a strangeness of +1, X must have a strangeness of -1 to conserve strangeness. I know strangeness can change by +/-1 in weak interactions, but as we know X has strangeness, it must have a strangeness of -1 to satisfy strangeness not changing, or changing by +/-1.
X must be a baryon to conserve baryon number and it must have a charge of +1e to conserve charge.
So, it therefore has three quarks, one being a strange quark, and the other two must be either up, charm or top quarks to conserve charge.
Why must it be two ups (if it even must be that)? If a quark changed flavour from an up to something else the weak force would be in play. Is that a possibility?
 A: As dmckee says in the comments, you have found a first order diagram.
What are Feynman diagrams? they are the diagrammatic  representations of complicated integrals that will give the cross section of an
interaction. These integrals are a series expansion , and each term has a diminishing constant allowing to use the first terms to first order, the way you have drawn. They are very useful because they simplify adding up quantum numbers that have to be accounted for.
It is the strong interaction because there is a gluon in first order, i.e the gauge particle of strong interactions. If you look at the ordering of interactions in terms of the coupling constant, you will find that a first order diagram with a photon exchange, which also creates quark antiquark , is suppressed by its coupling. The weak interaction is suppressed by its coupling even more.
Please note though that because of the coupling being 1 the series of perturbative diagrams for strong interactions do not converge and the first order diagram you have drawn cannot be calculated with perturbation expansion (There are innumerable diagrams with gluon exchanges in higher orders ).New mathematics are used depending on the calculation needed, dominant being QCD on the lattice. The diagram is useful for quantum number conservation display but not for calculating. 
