The rules for Mandatory Actions can be found on page 11 of the Supplement. They are fairly in depth, but here is an excerpt which I think answers this, and other similar questions.
A creature must perform a mandatory action if it is possible to do so when it chooses what action to perform. If the creature is under multiple mandatory action requirements, the creature may choose which one of the requirements to obey. These effects do not allow a creature to perform an action that would be otherwise forbidden, such as attacking a creature that would otherwise be an illegal target (for example attacking a creature when a different one is guarding, or attacking a Flying creature if the attacker is not flying).
Note that declaring the action includes paying any costs, so one would be required to pay any costs required to perform the action. Determining whether or not the condition is possible is checked only at the moment that the creature would be choosing its action (or target); the controller of the creature is not required to do anything beforehand that would make the action possible.
Basically with Mandatory Actions, you have to always ask yourself "Can I do this Mandatory Action" immediately before you take an action every time you go to take an action. If the answer is Yes, then you must do that Mandatory Action. If the answer is no, then you don't have to worry any further about the Mandatory Action. It doesn't force you to do anything else to make it possible to do it, and you can then completely ignore it for that action (checking again the next time you go to take an action).
With Rooted and Chant of Rage, you activate the creature and then ask yourself "Can I attack the creature with the Hate token?" If the answer is no, then you must check the secondary mandatory action by asking "Can I move closer to the creature with the Hate token?" If you are rooted then the answer is no. At this point, you are free of all obligation and can do whatever action you want. However, if you choose to pay the Uproot Cost, then the answer will be that yes you can move closer, so you will have to do so.
Suppression Orb is different. It applies a cost to move, which doesn't itself prevent you from moving. Uproot is a cost to remove the Rooted Trait, not a cost to move. It is a subtle difference, but an important one. Mandatory Actions do force you to pay the cost of doing the Mandatory Action itself, but since Uproot isn't a cost to move it is exempt. Chant of Rage doesn't say anything about having to remove the Rooted Trait, just about having to move. Does that make sense?
