In general, only if aesthetic only. If the edit would have an effect on gameplay, then it should not be modified. Also, the mod should indicate that they have edited a post both by indicating so in the edited post and by courtesy PM to the edited player. Though I suppose the exception would be a deleted doublepost, which ipso facto would have no post in which to indicate the modedit, though a courtesy e-mail would still be appropriate.
Specifically:
1) Remove/blank double-posts
Yes
2) Spellcheck
No
3) Complete/fix broken formatting tags
Yes. (Unless, as I stated it effects the game; ie, if a vote is unbolded, it should be left unbolded.)
4) Remove personal attacks
No. If it's that bad, remove the player, not the post.
5) Fix unintentional smilies
Yes
6a) Remove accidental outing of roles (for any game)
Only if the role outed is from a different game in progress. (ie, a player in game A outed a role in game B; A =/= B) This isn't your call; it's the call of the mod of game B, and you should take measures to minimize the level of damage control they might need to do. A swift modedit by you might save their game; a delay might kill it. Even if it seems too much time has passed, it is not your place to remove their option.
6b) Remove accidental outing of personal information (home phone, last name, SSN, etc)
Yes
7) Add missing formatting
Not sure I understand how is this different from (3)...
8) Anything they're asked to do in thread/via PM
I'm not sure what is being asked here, but anything is a dangerous word. So, no.