Is Earth's orbit stable enough not to be thrown away from the Sun after such a collision? Are there interactions between Earth and other large bodies which have a stabilizing effect? I know there's some interesting dynamics in the Earth-Venus interactions but I wonder if there's enough there that you could consider it stabilizing per se.
It depends entirely on the momentum of the impactor. If the impactor's momentum was much less than the Earth's then it would have little effect.
We hypothesize that Thela was like the size of Mars. Since all orbits in the solar system are due to gravity (assuming no rocket powered alien space rocks), we can calculate the maximum velocity a planetesimal would have as if it were falling from infinity into the sun. I can't be asked to do this right now, but based on the Virial Theorem and Earth's orbital velocity of 30 km/s, I bet ~60 km/s is the theoretical maximum impact speed. Mars has 10% of Earth's mass, so at best Thela would have 20% of the Earth's momentum.
And more likely the impact occurred in a grazing fashion in order to make the moon and with much less velocity, since it was probably already in an orbit around the sun. So back of the envelope, we wouldn't expect much to happen to Earth orbit other than a small disturbance that makes it more elliptical for a time. Then over the next few Gyr the solar tidal interactions circularize it.
The ocean is a tiny fraction of the planetary mass, so it's mostly the bulk mass of the planet. If you watch the moon forming simulation posted here, you can see right before impact the bulging of Theia toward Earth. This demonstrates the non-rigidity of planets pretty well. This happens to a much smaller extent between the moon and Earth and is the reason why the moon is tidally locked (one side always faces us) and is slowly moving away from us -- the tidal forces are transferring angular momentum from the Earth to the moon. Consequently as the moon moves farther away, the length of our day will increase.
