From the diagram, looking from the lower left hand corner, follow the arc up to the right and see how the Moon takes quarter revolutions around the Earth. So that is the way the Earth "throws the Moon up and over itself" but it is not just the Moon orbiting the Earth but the Moon is coming and going from the Sun as well and this will affect its kinetic energy to potential energy balance. So I guess it is really difficult to take the eccentricity out of the Moon-Earth system, even though with the Yo-yo Moon Capture theory when they were tidally locked, I'd like to know whether all this eccentricity was present then or not? If my reasoning is correct the eccentricity is inherent, but with the larger orbit, as the Moon is making now compared to then, I would hazard a guess that the eccentricity is getting worse because the radius is getting larger (the Earth is adding energy all the time), but on the other hand it could be lessening because the orbital period is getting longer. So which effect dominates?
Any comments on what the answer maybe please?

The study below has picked up an increase in the eccentricity of the Moon. So maybe the action of throwing the Moon into a higher orbit, which implies the Moon gets both closer to the Sun and further away depending on which side of the Earth it is on, has an intensifying effect on the Moon's eccentricity. Well that is my rough and ready guess.

The following study could be read:
"On the anomalous secular increase of the eccentricity of the orbit of the Moon"
http://arxiv.org/pdf/1102.0212.pdf

I thought it a bit odd they didn't have an explanation for the increase in eccentricity yet I had proposed a reason even before I had known about this scientific study!

How does a Theia collision alter the history of life on Earth? It wold be another 800 million years before the appearance of life on Earth, still ample time for development.

I want to clear up one thing, for there seems to be a contradiction in what I have said in the last several posts:
1. "3. At each pass of the planets Earth and Luna, the relative motion between the planet Luna (Moon) is slowed by speeding the Moon up (lower Lunar -Sun orbit)
(This must mean the period between conjunctions increase with time, as the orbital periods become synchronized. This bit is a logical consequence of #3.)"

And prior to that:
2. "The Moon gains speed on one leg, the Moon and the Earth move toward each other relatively (as well as orbiting the Sun, so I'm not suggesting anything goes backwards). The Moon would then slow on the going-away half of the orbit. I think that means the two halves are dynamically balanced."

So I will agree that these two statements are partially true but also a little contradictory too, for I would say that if there was angular momentum transferred the two halves of the orbit will not be fully dynamically balanced as a consequence. The transfer of the momentum dampens the balance.
This transfer of momentum would tend to make the Earth and Lunar spin anticlockwise looking down on the system. [I will try to use the term Luna for the planet Luna prior to its capture as the Moon.]

If the co-planet Moon is pulled toward the Earth how much does it speed up? does the speeding up significantly change the period of conjunction? I'm thinking it must. 16058 years is time taken for the natural difference in orbital period of the Early Moon-Early Earth to allow one complete lap, if they accelerate toward each other as they approach, some of this is reversed as it leaves the Earth, so that must mean the average velocity is higher and period is reduced.
But by how much? The Moon gains speed on one leg, the Moon and the Earth move toward each other relatively (as well as orbiting the Sun, so I'm not suggesting anything goes backwards). The Moon would then slow on the going-away half of the orbit. I think that means the two halves are dynamically balanced.
With our calculations we will use both the Early Moon-Early Earth as 42 times the current masses, remember. So you might agree the Earth and Moon at opposite sides of the Sun would have only minimal gravitational affect on each other. At the conjunction the tug will be the strongest. I know this isn't rocket science.

Assuming it is a circular orbit there is a radius and the change in "theta" the angle between them.When the speed changes both theta and radius will change too.
Since the Moon is much less massive than the moon we could look at the situation assuming all the changes affect the Moon only to begin with and when we feel happy with the results we will complicate the situation by having them both move.
Earth:Moon mass ratio = 81.3:1 (Moon Earth mass ratio = 0.0123)

Theta change per year = 360/16057
0.022420128
so we will look at 10 year blocks. This gives 0.22420128 degrees change in theta every 10 years and we will look at this finer when the rates become significant. There is probably more movement occurring in the last year prior to conjunction than in the first 5000 years after their greatest separation. So sensitivity in the later stages will be important.

Probably a bit like a double pendulum? - Just joking. Try and predict the motion!