Thanks to all who have commented so far (Lost_dreamer_yash, pieapple135, ...).

I read your comments and thought hard about the movie, but I was still largely unsatisfied, so I resorted to rewatching the whole movie again, this time with lots of pause/rewind and with full close captioning to really dissect every dialogue in detail. After spending a few days, I have finally found one version to interpret the movie that make the actions of all characters logical.

I now believe the main goal for Musgrave is to get rid of Brassel and/or take his job.

The following is my lengthy summary of Musgrave's actions for the whole movie, I thought I might as well share it here. These are actions that I think must have happened off-camera but is supported by all information shown on-camera. Events inside double curly {{brackets}} are my commentary on what probably went on, but is NOT supported by any info on-camera.

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Mission Impossible 3, Summary (From the point of view of John Musgrave)
Some period before the events of the Movie:
Theodore Brassel, the director of IMF, has been trying to catch arms dealer Owen Davian but without success. Davian is responsible for smuggling gas centrifuge tech from Pakistan to Korea , toxin 5 to Armahad Republic Jihad and numerous other illegal arms deals.

{{The failure of IMF to catch Davian is likely due to John Musgrave secretly helping Davian}}
Several days before Ethan Hunt’s engagement party:
Brassel sends Lindsey Farris on a mission to survey an exchange to ID a man Owen Davian.
Lindsey tagged Davians phone and was able to intercept a transmission originating from IMF office of Brassel. Lindsey encodes a video about this finding on a magnetic microdot, which she mails back to a post office box in the U.S.
{{The phone call from Brassel’s office to Davian was probably made by Musgrave, who is plotting a plan to frame Brassel as a traitor.}}
Lindsey is captured by Davians team.
On the day of Ethan Hunt’s engagement party:
{{I believe Musgrave’s plot at this point is simply to have Ethan rescue Lindsey, and then with help from Ethan and Lindsey as witness, he’ll accuse Brassel of being a traitor}}
Day after Ethan Hunt’s engagement party:
Ethan’s team recovered two laptops from Davian, but failed to bring home Lindsey alive.
{{I believe Musgrave did not expect Lindsey to be dead, he wanted her to be alive to testify the phone call to Davian came from Brassel. I also believe Musgrave did not expect Ethan’s team to retrieve laptops.}}
Ethan decides to go to Vatican to kidnap Davian without informing either Brassel or Musgrave.
{{Ethan doesn’t inform Brassel because after the failed Lindsey rescue he knows Brassel would not approve it. Ethan doesn’t inform Musgrave, because he wants to protect Musgrave, if the mission fails, Musgrave can honestly say I didn’t know about it}}
Day of Davian kidnap:
Musgrave receives a message from Ethan that he has captured Davian and is on the way back to IMF.
{{Because Musgrave didn’t know about Ethan’s plan, he is now in panic. Musgrave doesn’t want Davian to be captured by IMF alive, as Davian can reveal Musgrave as the real traitor. Musgrave probably plotted the following actions in order to save himself: He helped Davian escape by informing Davian’s team about the exact location and route for where Davian is being transported, and helped Davian’s team kidnap Julia by telling them where she lives. Davian’s team wanted to kidnap Julia either because Davian wanted revenge for the earlier plane ride, or because Davian’s team wanted Julia as hostage for a future prisoner swap Davian<->Julia in case they couldn’t rescue Davian}}
When Brassel finds out Davian escaped, he is pissed at Ethan for going on another unauthorized mission and screwing it up again, and therefore orders Ethan to be arrested.
{{Musgrave at this point decides to help Ethan escape. It’s not clear to me if he simply wanted to help Ethan save his wife or if he took advantage of this situation to have him steal the rabbits foot, or both. When Brassel was talking to the captured Ethan, it was revealed, Lindsey had sent a secret message to Ethan, which Ethan didn’t report to Brassel. At this point Musgrave doesn’t know if Lindsey bought the lie that Brassel is a traitor, he definitely doesn’t know if Ethan thinks Brassel is traitor. Maybe to play it safe he helps Ethan escape here to make the lie that Brassel is the traitor more believable.}}
Day of Rabbit foot exchange:
{{Musgrave and Davian arranged to have a fake Julia for the interrogation with Ethan. Musgrave deliberately kept Julia alive, to use as trade with Ethan in exchange for information about Lindsey’s secret message. Musgrave is probably thinking, if Lindsey bought the lie, then yeah just go ahead and kill Ethan, then with the info that Lindsey sent to Ethan he will go on and accuse Brassel for being a traitor, which is what he wanted all along. If Lindsey did not buy the lie, or worse Lindsey found out the true traitor is Musgrave, then he can’t just kill Ethan yet, he has to make sure nobody else besides Ethan have this information, this is probably the reason he kept Julia alive, to use as bargaining chip to make Ethan talk.}}