Read the rules. Often you can work out what they were likely was removed for. However, I do think the deleted notification should be changed slightly, and a link to the rules provided as well, so again, people can try figure out why their post might have been removed. Most of the time users just aren't aware of the rules and the post may appear legit to the person who made it, but could have been a duplicate, double post, or in a necrobumped thread etc.
And the user should be made aware of the rules he/she is supposedly breaking.
An option for staff to leave a note or choose from a drop-down menu of the most common reasons would be helpful. Ie "double post", "off topic", "no ref links" etc.
Nah. Most users won't do it and if it's against the rules then staff should just remove it.
It should be anonymous. Staff members would just be hassled if it told you who deleted it. I sometimes even get angry messages asking why a post was removed even though it had nothing to do with me but users assume it was me because my username is on that sub-board. Thank yourself you even get a notification though. On many other forums you don't and your post just disappears with you being completely unaware.
As others have said, it it was just a 'thanks' then that maybe why.