I was recently reading through the rulebook and had a couple of things I need clarifying:
1) Does the automatic fail of a courage test for a riderless horse happen only after its rider dies or does it happen whenever the rider is thrown?
2) Are spearmen/pikemen also pushed back when the person they are supporting loses a fight?
3) Is there a bow limit for the Haradhrim in the Howdah?

1) Whenever it no longer has a rider on its back for whatever reason.

2) Not necessarily. A supporting model can (and often will) be the one model that is allowed to make way for a figure retreating from a lost duel. If the fighting figure can already move away sideways, the supports can stay in place as well. As only a single model can make way, pikemen in the second supporting rank can actually result in having the fighting figure being trapped (if it can't escape sideways, or if the first support can't move away allowing the fighter to retreat).

3) They are part of your force and follow the normal restrictions - assuming you want all figures in the howdah to have bows, you will need twice that number as additional troops on the ground to not exceed the bow limit (or the same number of additional troops if using Harad's special 50% bow limit).

Thanks!