The answer to (a) is yes. Consider the LP in the image below, whose feasible region is the triangle ABC. The decision variables are $x$ and $y$, and the slack/surplus variables are $s_1$ and $s_2$.
The simplex method starts at A and (after pivoting an artificial variable out of the basis) has $x$ and $s_2$ as the basic variables. Since the rate of increase of $f$ is faster along edge AB than along edge AC, the next pivot moves from A to B, removing $s_2$ from the basis and adding $y$. The final pivot moves from B to the optimal vertex C, removing $y$ from the basis and adding $s_1$.
The answer to (b) is trickier. I don't think this can happen at a degenerate corner, and it cannot happen at a nondegenerate corner. I'll just give a hint to the proof in the absence of degeneracy: if it did happen, you would be moving along an edge of the feasible region that contained three distinct vertices, which contradicts the definition of a vertex.