If two particles cross, then aren't they in the same position? To me, this answer doesn't seem to make sense; if the paths cross twice, then how come P and Q are never in the same position?

 A: It isn't meant to read "if the paths cross twice then p and q are never in the same position (at the same time)."  It is certainly possible that they could be in the same position at some time if we didn't know anything else.
BUT we do know more.  We know the full equations for where the particles are at any given time.  We also know that whenever two particles are in the same position, it must be at a place where paths cross.  However, at neither of these locations will both particles be there at the same time.
$p$ will be at the origin at time $2$, whereas $q$ will be at the origin at time $3$.  Similarly for the other point of intersection.  Since for neither of the two places where the paths cross are the particles there simultaneously, we can conclude that the particles never collide.
A: HINT:
Consider the following: I live on the $10$th floor of a building and every day at $10$ a.m. I leave for work and use the elevator to go down to the first floor. I return home at $8$ p.m. following the same path.
My neighbour, who lives right next to me (also on the $10$th floor) goes to work at $11$ a.m. and works on the other side of town. He also leaves for work using the elevator. He returns home at $10$ p.m.
Are we, at some point in time, in the same position?
