How to display the user edit-picture form in custom block?
The ID you pulled from the DOM isn't the form ID, unfortunately. It's just the ID of the user image picker fieldset.

Take a look at the User Picture Field module. I haven't used it, so I can't personally vouch for it. However, it does make the user picture a field, which adds a lot of flexibility. I think Panels will let you move the field to the place where you want it to appear. You should also be able to use the editablefields module to make the image editable in the display.