Could the singularity that gave rise to our universe be past-eternal? Are there any compelling models that include a past eternal singularity that ultimately gave rise to our universe? Does the "no-boundary" hypothesis that utilizes imaginary time have past eternal imaginary time and real time being created at the big bang? What are the underlying aspects of physics that allow or disallow different past eternal models of the universe? Are these questions even really answerable without a complete theory of quantum gravity? Your answers to my hopefully somewhat comprehensible questions are extremely appreciated.
 A: The Hartle-Hawking no-boundary proposal suggests that the universe as we know it was not created from nothing.  Rather, its history would be finite, though unbounded.
In the words of Stephen Hawking:
"The no boundary proposal, predicts that the universe would start at a single point, like the North Pole of the Earth. But this point wouldn't be a singularity, like the Big Bang. Instead, it would be an ordinary point of space and time, like the North Pole is an ordinary point on the Earth, or so I'm told. I have not been there myself.
"According to the no boundary proposal, the universe would have expanded in a smooth way from a single point. As it expanded, it would have borrowed energy from the gravitational field, to create matter. As any economist could have predicted, the result of all that borrowing, was inflation. The universe expanded and borrowed at an ever-increasing rate. Fortunately, the debt of gravitational energy will not have to be repaid until the end of the universe."  - The Beginning of Time.
It may be misleading for you to call this proposal "past-eternal".  Time, as we know it, would have begun at a space-like point where prior time could be conceived sort of like the i in a complex number, as far as I can understand it.
The problem with the Hartle-Hawking proposal is that it requires a closed universe.  However, observations indicate that there may be insufficient matter in the universe to stop expansion, and this would result in an open universe.  This limitation of the Hartle-Hawking proposal is discussed in a website prepared by Jack Wong at the University of Victoria in 1999.  I could not find any information about the author, but if you scroll to the bottom and click on Table of Contents you will find that the website addresses some of the ideas you asked about. 
In Sean Carroll's book From Eternity to Here, he explores various ways that the nature of time, which we conceive of as an arrow from past to future conforming with the 2nd Law of Thermodynamics, could have come about.  He says that one way of dealing with the arrow of time is simply to posit that the universe had low entropy at a certain point near its beginning.  This would set a boundary condition, like a law of Physics, that the early universe had low entropy, and that's that.
He mentions the Hartle-Hawking proposal in footnote 275, where he seems to find the no-boundary condition not necessarily opposed to a boundary condition: "A related strategy is to posit a particular form for the wave function of the universe, as advocated by James Hartle and Stephen Hawking (1983).  They rely on a technique known as "Euclidean quantum gravity". . .  It has been suggested that the Hartle-Hawking wave function implies that the universe must be smooth near the Big Bang, which would help explain the arrow of time (Halliwell and Hawking, 1985), but the domain of validity of the approximations used to derive this result is a bit unclear.  My own suspicion is that the Hartle-Hawking wave function predicts that we should live in empty de Sitter space, just as a straightforward contemplation of entropy would lead us to expect."
A: The possibility suggested by the OP's title question can actually be traced back to ancient Hindu philosophy, where it was known as "the cosmic egg", and is considered to have influenced Lemaitre (the "L" in the FLRW model).  As discussed in the paper by Alexander Vilenkin and Audrey Mithani that's freely visible at https://arxiv.org/abs/1204.4658, it's not currently considered viable, as the "egg" would have been susceptible to quantum fluctuations that would've prevented its completion of a trajectory from past eternality.
A: Today, most physicist "believe" in the inflationary cosmology which solve many problems in cosmology and it doesn't indicate our universe started with a singularity.
