do T cells proliferate as soon as they come into contact with an APC or do they find a B cell first (is this costimulation?)
They don't proliferate "as soon as" they meet their Ag but they don't need to find a B cell first. Costimulation is just signals provided by an APC to trigger proliferation.
how do cytotoxic T cells proliferate if they don't come into contact with antigen?
What makes you think they don't...? How else would they be activated?
Is the diagram inaccurate? Compound R almost appears to be an enzyme catalysising the reaction of Q into S, but the answers suggest Compound R can be turned INTO S as in an alternative metabolc pathway, is the diagram incorrect, or is this always how metabolic pathways are drawn?
Diagram is accurate. Compound R is not an enzyme, it's a compound. It can be converted to S, just like Q can (but presumably a different mechanism).
Enzymes are written next to arrows, they don't get their own box.