It depends on the reversibility of the reactions, relative rates, and the conditions you conduct the reaction under.
if both reactions are reversible And if you conduct the reaction under reversible conditions (i.e higher heat, longer reaction time) then the more thermodynamically favorable situation will prevail.
Another way you can look at it is from the perspective of kinetics. If you are under kinetic/irreversible conditions then whichever reaction is faster will prevail.
All this is to say since you are forming the same product from two different reactions, under reverisble/thermodynamic conditions, the most thermodynamically stable species will be most prevalent so that can be A B or C depending on what their Gibbs free energies are.
To answer your question plainly, it doesnt mean C wont react with B; you need to assess the relative rates of reaction to determine whether the reaction will happen in realistic amount of time or not.
Very plain and simply, no. In fact, just because conditions favour better the reaction of B with A, the reaction of C with A might even go faster. A well-known example of this is the oxidation of iron and zink. Under normal conditions, Gibbs free energy balance tells us that the oxidation of zink is more prone to occur than oxidation of iron, however the kinetics of oxidation of iron drive the reaction to occur much faster.