Orbital transfers and escape velocity: relationship

[brainstorm]

I have been told that there is no interaction between the gravitational fields of the sun and any planet, but the issue continues to stimulate my curiosity.

If the Earth orbits the sun at 30km/s, how could anything achieve an orbital speed around Earth at greater than 30km/s without transferring into solar orbit?

If it is the case that solar gravity influences escape velocity, then wouldn't something orbiting one of the distant gas giants be able to orbit those planets at a much more distant point at a lower velocity without escaping into solar orbit?

In that case, wouldn't a planet's escape velocity depend on its orbital velocity relative to the sun or whatever it was orbiting?

 

[The riddler]

"I have been told that there is no interaction between the gravitational fields of the sun and any planet"

Could you please explain what you mean by this

 

[brainstorm]

"I have been told that there is no interaction between the gravitational fields of the sun and any planet"

Could you please explain what you mean by this

Ok, maybe this example is simple enough to clarify what I'm talking/asking about:

Imagine there are two massive bodies with the mass of the sun orbiting each other at close range. Let's say they have the same distance as Jupiter does from the sun. At such close proximity, these two bodies would have to orbit each other at a very high velocity to maintain constant distance from each other, correct?

As such, any object that came near the system of the two bodies would have to maintain a level of velocity that matched to some extent that of the two bodies' orbital speed. If the object was going much slower, it would fall into one of the two bodies and if it was much faster it would slingshot by them and continue on.

Now, if one of those two bodies would be alone, i.e. without orbiting the other body, it would have the same escape velocity as the sun since it has the same mass, right? That means that if an object was traveling at Earth's orbit, it would be moving at 30km/s, like Earth, yet still be far from reaching escape velocity of the solar system.

Yet, if the two sun-mass bodies are orbiting each other, an object cannot ascend much beyond Earth's orbit without falling into orbit around the second body, correct? So the escape velocity of the first body with the sun's mass was reduced by the fact that another gravitational field overtook the first one prior to the object reaching "escape velocity."

So, what I think (but don't sufficiently understand, which is why I'm asking about it) is that gravitational fields interact to limit how much velocity a satellite within one gravitational field can achieve before transferring orbit into another gravitational field. If that is the case, then I think "escape velocity" is an inadequate concept in that it treats each gravitational field as if it exists independently of others. In fact, I think the entire universe is composed of contingent gravitational fields, however weak they may become at their shared boundaries.

Now, if you thought critically about the idea of distant gravitational fields having a shared boundary, then hopefully you realized that one object's gravity doesn't necessarily give up just because another object's gravitation overshadows it. Take the effect of the moon on the tides. The moon's gravitation does not stop at the point Earth's gravitation becomes stronger. So, theoretically, any object traveling anywhere between the Earth and the moon is interacting with both gravitational fields simultaneously, right?

What interest me is how such gravitational field interactions may affect space/time dilation and the behavior of matter and energy generally.