... [The President] may, on extraordinary occasions, convene both Houses, or either of them ...
The word "only" is not there. The Speaker can call the House into session anytime.
I don't think you are correct on that. Article I, Section IV, Clause 5 says that consent of both houses is needed to adjourn for more than three days. It should follow that consent of both would be needed to cut the adjournment short. Neither Boehner or Reid can do it on their own.
The speaker Waldo Boehner can also call a press conference to
demand a vote on a strike. The fact he as not do so tells you everything
you need to know about Waldo. He is in on it with Obama.
Where in the world is Waldo Boehner?