Can I remove some of the strange attributes from a strange cosmetic?

For most cosmetics, @EnragedTanker's answer would be correct.

Strange Haunted items are special. Why? Because they have the following intrinsic properties/parts that can't be removed:

• Strange Filter: Carnival of Carnage
• Strange Part: Carnival Underworld Kills
• Strange Part: Carnival Games Won

Why? Because they were part of the Halloween 2014 event, which introduced sd_doomsday_event AKA Carnival of Carnage.

Yes.

Strange Parts can be individually removed from a weapon using the Restore button in the backpack, allowing the player to remove a strange part of their choosing; however, a part which has been removed will not be returned to the player, and instead will be permanently lost.

-- Strange Part, Team Fortress 2 Wiki

Right click on the strange item in your backpack, click Restore, choose the part you want to remove.