If a reactant appears on both sides of an elementary equation, can it be cancelled?
If the elementary step is indeed written 

$$\ce{A + B -&gt; A + C}$$

and assuming that you haven't just written in $\ce{A}$ for fun, i.e. $\ce{A}$ is actually a participant in the step, then yes, this would be considered bimolecular. This sort of step, featuring the same chemical species on both sides of the reaction, does pop up in (for example) the Lindemann mechanism.

With regard to kinetics, however, it is worth noting that steps of this kind can sometimes indicate a reaction that is catalytic with respect to $\ce{A}$. If that is the case, then the concentration of $\ce{A}$ is essentially constant and you would obtain pseudo-first-order kinetics.