Learn English – What grammar structure is “having + [verb]-ed”?

Here are the three cases you have presented:

(1) I gave Susie a picture of herself.

(2) I went there by myself.

(3) Do it yourself.

I have come up with a new rule under which your three cases as well as the traditional rule are subsumed:

A personal pronoun must be in the form of a reflexive pronoun in order to refer back to another word when both the word and the personal pronoun are contained within a single clause.

I'll call this new rule "the single-clause rule". The single-clause rule surely incorporates the traditional "subject-object" rule, because a subject and an object are contained in a single clause.

Now, let's see if the single-clause rule explains the three cases above.

Regarding (1) and (2), herself and myself refer back to Susie and I, respectively. In (3), yourself refers back to the implied subject you. And these are all in a single clause.

There's two different constructs that match what you're talking about.

1. A present participle giving more information about something that's happened, as in the comment example: "John, having the gun, ensured that nobody left the room." This describes John by mentioning that he has the gun.
2. A gerund, treating an action as a noun. An example would be "John having a gun made the situation ten times more explosive." Here, the phrase is being used to mean "The fact that John had a gun." and that implicit fact is the subject of the verb made.

Useful further reading on these two: http://www.chompchomp.com/terms/gerundphrase.htm