I have a driving test coming up soon, but I'm unclear on what the examiner will expect me to do on a red light right turn through a dedicated right turn lane. This is the sort of thing I'm talking about:

For red light right turns on "normal" intersections, we come to a stop behind the crosswalk, look both ways and complete the turn when the way is clear. If we're doing a red light right turn through a dedicated right turn lane (as shown above), are we required to come to a complete stop AT ALL TIMES even if the way is clear? If so, do we stop before the crosswalk or after the crosswalk?

When you approach the sign you have already turned and are no longer facing a red light so a full stop is not required unless you have to yield. In your diagram specifically you only have to yield if necessary (if there are pedestrians crossing you would yield to them and you would then yield to oncoming traffic before you merge).

If we have to yield to other cars, do we stop before or after the crosswalk?

First you check for pedestrians if clear pass the crosswalk then before merging check for cars, if clear merge if not stop and wait. You would normally stop before the curb line if you had to wait for cars to pass. Stopping too far back may unsafe as you will not be able to see oncoming traffic.