Are a rider and mount considered different targets so I could push or pull one but not the other with a power?
If I can push the rider off of the mount, would the "Catching yourself"(PHB p284) clause kick in and give the rider a save from the fall?
Essentially no, they both move together even if you only target one of them.
From the Compendium Also pg254 in the Rules Compendium
The mount and rider are considered separate creatures and are targeted independently. (RC p254)
The attack sequence if multiple targets are involved is on pg214 of the Rules Compendium. Simplified it is:
In the case of a mount and rider you should fully resolve the attack against one before going onto the other. If the mount is slid left and the rider slid right, they will likely end up right back where they started.
In the case of teleportation, (which is not forced movement). It is covered on pg255 of the Rules Compendium. In short they teleport separately, teleport the mount and the rider is dismounted and falls prone.
Based on current RAW, the answers aren't quite as simple as the questions - in fact, part of them brings up another question which I'll be shortly sending to WotC support.
First, let's break up your first question a bit:
Yes. For the purposes of targeting, the rider and mount are separate entities. If you have a power that targets one creature, you must choose either the mount or the rider. If you have a power that targets all creatures within a burst or blast that includes the mount & rider's space (remember, they share the same space) then you must roll separate attacks for each.
For a push, pull, or slide:
For a teleport (technically not "forced movement", but can be involuntary):
I'm certain the answer to this would be "no". This being because, in most cases, (push, pull, slide) a rider's separation from the mount due to forced movement is the rider's choice - it either happens or it doesn't, no save involved.
In the case of a teleport, there is only one situation in which the rider would explicitly be dismounted and fall prone. This is when the mount is the target of the teleport. Since the mount is the creature being affected, and rules specifically state that rider and mount are not teleported together, the rider has no choice in the matter - he dismounts and falls prone.
In any case, the "Catching Yourself" clause would not even be effective for forced dismounts, since a successful save still leaves the target prone in the last space occupied prior to the fall.
Some scenarios not explicitly covered in RAW:
After reading the rules on this one more closely, I see it is a direct RAW interpretation. The key lies in this bit:
In this case, the condition being triggered is that the rider is being moved, and the mount is also. This overrides the "mount carries rider" clause, and violates the "rider chooses whether they move together" clause. Therefore the two are separated, the rider is dismounted, and falls prone.
All of the data to support my conclusions thus far, are taken from WotC Customer Support e-mails, and the following Rules Compendium sections:
If only one of the two (mount and rider) is subjected to forced movement then generally no you cannot force a dismount (as explainied by Pat). But imho that doesn't fully address the more general question (Can forced movement be used to dismount a rider).
If you are using an attack that can target more than one creature, it gets more complicated (mount and rider are targeted separately and therefore bursts and blasts can affefct both of them). Certainly it can be argued that the rule states that if the mount is subject to forced movement then the rider can always simply ignore any competing forced movement - although it's not clear that is the intent. Likewise, what happens when the mount is immobilized and the rider is subject to forced movement - the rider can't have the two of them move together - is he then dismounted?
What makes sense to me (and I leave it to each DM to decide to what degree in-game logic versus strict and literal interpretation of the RAW impacts their rule decisions) is the following: If the mount is immobilized, forced movement can force the rider off of the mount. If the rider is immobilized, it can ride along with the mount as it is moved. If both are subject to forced movement and moved in opposing directions the rider can be dismounted.
It may not be strict RAW, but the rules are not and cannot be 100% comprehensive and I feel this specific sub-question is not explicitly covered by the RAW.
I asked this question on the D&D forum, check it out here
The part of the rule in particular that I'm asking about is:
I believe that you can dismount a rider, when and because the rider "cannot" move her mount all the way. After a while people see my point of view.