In the end of Star Wars: The Clone Wars (season 5), Jedi order was told to kick Ashoka out of order before she could be dragged to the court. It looks to me that Republican laws couldn't put someone from Jedi order on trial. Or maybe, it was a conventional respect to the Jedi order.
However, what would happen if Jedi order rejected this proposal of kicking out Ashoka? Other than Order 66, what other civil solutions The Galactic Republic had against the Jedi order? Anything from EU?
Generally speaking, members of the Jedi Order speak as if they were susceptible to prosecution for crimes. For instance, when the Obi-Wan suggests that Anakin spy on Palpatine, Anakin suggests that doing so would be treasonous (and thus a crime):
OBI-WAN: (takes a deep breath) The Council wants you to report on all of the Chancellor’s dealings. They want to know what he’s up to.
ANAKIN: They want me to spy on the Chancellor? That’s treason!
OBI-WAN: We are at war, Anakin. The Jedi Council is sworn to uphold the principles of the Republic, even if the Chancellor does not.
Revenge of the Sith
That the Jedi can commit treason is also suggested by Palpatine:
MACE: The Senate will decide your fate.
PALPATINE: I am the Senate!
MACE: Not yet!
PALPATINE: It’s treason, then.
Revenge of the Sith
This also shows that even the Supreme Chancellor is subject to the laws of the Republic (at least in theory), which strongly suggests that the somewhat extra-governmental force that is the Jedi would not be immune.
Obviously, members of the Jedi Order would not be likely to be prosecuted for crimes (especially minor crimes such as property destruction), particularly if they were committed in the course of apprehending criminals or defending the Republic from the Separatists, but they don’t seem to be above the law.
From what I know, the Jedi Order never had any special rights or immunity from anything. They were highly respected everywhere and partially feared, but they were far from being immune.
This is also visible at the end of Episode 3: Order 66 is issued, but at the same time the Jedi are declared traitors or enemies of the republic for those that need an explanation as to why such a respected order is attacked/killed at once (i.e. for the public).
Order 66 isn't part of any special law or something like that either. It's more like a "trigger command" immediately flipping a switch in the clones' minds, so they attack/betray the Jedi without asking questions or declining (e.g. due to being friends).
As for the Clone Wars episode in the question: I haven't seen it yet (only first two or three seasons I think), but I guess this would be just to follow the law etc. without causing negative impact on the order's credibility.
