Questions like these, deleting activity without deleting all posts one by one, are being asked regularly, and Facebook does not offer an automated solution for it. (I suppose they have their reasons not to.)
There are third-party scripts that will allow you to, such as the Social Book Post Manager Chrome extension. I have tested it with moderate results. ...
This kind of situation comes when you clicked on a rogue app link. A lot of times things that are designed to make you want to click them end up installing an app on your Facebook account that will send out stuff.
Here what Facebook said about how to deal with spam:
Spam is sometimes spread through clicking on bad links or installing malicious software. ...
This Facebook KB article shows what a Story Impression is (emphasis mine):
Impressions are the number of times a post from your Page is displayed, whether the post is clicked or not. People may see multiple impressions of the same post. For example, someone might see a Page update in News Feed once, and then a second time if their friend shares it.
Facebook Pages are public, subject to any country and age restrictions set by the Page administrators. There's no way to hide your activity on public Pages administered by others, from casual viewing or from Graph Search.
For Facebook Groups, it depends on how the Group was created:
If you use an instant messaging client application, such as Trillian, Pidgin, Digsby etc., you can set them up to be able to chat over Facebook too, and those programs have settings that let you play a specific notification sound when a particular friend comes online.
The Wolfram|Alpha Personal Analytics for Facebook meets some of the features you are looking for, namely:
If you have friends in common/how many of them.
If you have common networks/how many of them.
It can list your friends by rank, depending on how many common friends you have with them. It can also show "mutual friend clusters", which are ...
From a technological perspective, the important problem which you are addressing -- that of a disgruntled admin "hijacking" a Facebook application -- does not seem to currently have a solution.
This problem is virtually identical to that of a Facebook page being similarly hijacked, and as mentioned in this article on Mashable from Sept. 2011, the latter ...