Does a luffing boat at the start have rights ? And,  Could another boat establish overlap on the luffing boat  ?

All boats have certain rights.
When Green establishes an overlap close to leeward of Yellow, Green must give Yellow room and opportunity to keep clear.
Yellow need not do anything until the overlap is established.
Therefore, it is risky business for Green.

A boat must keep clear of boats that have right of way, whether they are luffing or not. When boats acquire right of way, they must initially give a keep clear boat room to keep clear (rule 15). In the image you posted, Cyan has an overlap to leeward of Yellow.  The other boats are clear astern and must keep clear of Yellow.  If they establish an overlap to leeward and alter course, they must give Yellow room to keep clear (rule 16.1).  Rule 17 would apply after the starting signal, since there is no proper course before then and they are less than 2 boat lengths to leeward. They still must give Yellow room to keep clear - but Yellow has to do something - sheet in, head up, etc. to attempt to keep clear.  They can't just sit there and do nothing. 

Yellow is on starboard tack so any boats on port must keep clear of her. Boats clear astern of Yellow must also keep clear. 

Yellow must keep clear of any boats that establish an overlap to leeward of her. If a boat on starboard establishes an overlap to windward she must keep clear of Yellow. 

That doesn't change even if Yellow begins drifting backwards while luffing, but if Yellow backs a sail to move astern she must keep clear of all other boats. 

 They can't just sit there and do nothing.

That includes getting forced over the line early once the overlap is established ? Maybe even by green ?

And the pin end looks favoured, she should not be afraid to sail down there...

Yes, that does include being forced over the line.  It happens all the time.  Green is initially limited by rule 15 and then by 16.1,  Green cannot establish an overlap close to leeward and come up suddenly - they must give Yellow room to keep clear (and Yellow need not do anything until the overlap is established).

Thank you all. This will clear up a misunderstanding that another skipper was having.

|| "Yes, that does include being forced over the line.  It happens all the time.  Green is initially limited by rule 15 and then by 16.1,  Green cannot establish an overlap close to leeward and come up suddenly - they must give Yellow room to
|| keep clear (and Yellow need not do anything until the overlap is established)."

This is indeed interpreted wrong by many - thinking you can simply jump on a boat like "Jack in the Box" and claim rights ! Even when haling when still astern ... first overlap, then R11 cones into effect and R15 applies. With bigger yachts when jumping up close at the stern one cannot move to windward since your stern will kick-in/swerve while luffing.