If one wakes early and feels like he will fall back asleep, says the morning brachos and full Shema so he won’t miss its time, then does fall back asleep and wakes up again with JUST enough time to lay tefillin and daven Amida before sof zman ha-tfila, what is the proper order to complete the rest of his davening following his Shmoneh Esreh? (This is really a question with a few parts, or several interrelated questions, so I’ll try to be organized.)
(QP1) Should he say Tachanun to connect it to his Shmoneh Esreh? (I remember learning that Shmoneh Esreh and Tachnun are like two parts of one prayer.)
(QP2) Then after Amidah (and maybe Tachnun) should he:
(A) immediately return to Pesukei Dezimra, the Shema and its brachos, then skip the Amidah (and maybe Tachnun), say Ashrei, La-menatseakh and U-Va Le-Tsiyon, etc.
(B) continue as usual after Shmoneh Esreh all the way through Aleinu before returning to Pesukei Dezimra since he has already fulfilled the obligation to say Shema?
(QP3) Are the Pesukei Dezimra strictly required, or is a fatigued or distracted man permitted to skip them?
Before getting to the answer to your questions, the person should not be going back to sleep if there is a chance r fear that he will miss the proper zman. But in retrospect if he already saiud shema and brachos and then he only had tie to daven shemona esrei before the zman, he should say tachanun right after shemona esrei, because tachanun is in a way part of shemona esrei and shouls be said adjacent to it. After that he should go back to the first part of davening that he didn’t say yet.