The answer is that the reference "recordid" needs to have "I" capitalized (case sensitive):
So, yes, you CAN pass certain information to the Force.com sidebar menu.
I unfortunately have no clue where this recordId reference comes from or is documented. I am opening another question about that as I've investigated, and find no help online, and the source of the page has references "like this" but the other variables found there do not work (see below). At least "recordId" works for passing the current Id being viewed into the URL on the Force.com menu.
UPDATE: Also tried "merge field" references from Custom Button editor, replcing objectname with "record" (believing maybe, like "objectname.Id" that this is the "format). This did not work for other "merge fields".
For reference, the page source being tested:
Link to NEW question: What "References" To Fields/Variables Are Available To The Force.com Menu?