What happens if an object stops orbiting the sun? What would happen if an object (like a space ship) was to stop its orbit around the sun. would time slow down with that object (the same way time sped up for the characters in that great film Interstellar). Or even more so, if that object was to stop its orbit around the galaxy so to speak. And so on and so on. At what point is an object considered motionless? Could it be that the finite origin of the big bang has no time?
 A: One has to keep in mind why the word "relativity" was coined . It was needed because observations in physics tell us that there exists no absolute frame of reference , no absolute coordinate system. One is free to define coordinate systems  and explore the motions relative to the assumptions as long as Lorentz invariance applies strictly. 

What would happen if an object (like a space ship) was to stop its orbit around the sun. 

Newtonian physics would take over and the object will fall into the sun with a calculable trajectory.

would time slow down with that object (the same way time sped up for the characters in that great film Interstellar).

The clocks of the space ship would keep ticking and follow the Newtonian predictions until impact. Time does not speed up for the characters in the space ship, it is other reference frames that may see any slight difference because the velocity reached before impact would probably not be large enough for relativistic effects. 

Or even more so, if that object was to stop its orbit around the galaxy so to speak. And so on and so on.

Same answer. If General Relativity comes in because of proximity to enormous masses the effect gets more complicated but the answer is similar.

At what point is an object considered motionless?

When the reference frame is taken as the center of mass of the object.( In the case of the sun,above, the center of mass of the sun was assumed, and a relative velocity zero for a delta(t), then newtonian forces take over).

Could it be that the finite origin of the big bang has no time?

????  This has no meaning.
