Question 1: Is there a velocity, (relative to Earth), that a person/object could travel so that it is objectively ‘standing still’ against space?
No, there isn’t, but there is a “special” frame of reference if you need one:
Notice that most people think that you cannot define your motion relative to the C.M.B., which is false.
Question 2: What is the difference between a clock on the ship versus one on Earth, assuming they were synchronized when the ship left?
Note: this answer (assuming it is correct) is valid for every frame of reference. You don’t need to “stay still” or find a particular frame of reference.
When considering only speed, two observers in motion relative to each other will see the other’s clock going slower. You cannot say if you are moving, or the other or both.
Things change if you accelerate: if you go to Alpha Centauri, or if you reduce your speed to 0 relative to the CMB, your clock will slow down form the point of view of a person standing in the Earth. So, when you come back, you will be younger than your twin brother. This is called the twin paradox . An explanation of the paradox can be found in this video: https://www.youtube.com/watch?v=0iJZ_QGMLD0
Speed is relative, acceleration it is not. Closed in a science laboratory you cannot tell if you are moving at constant speed or not, but you can tell if you are accelerating (using a spring, for example). 0 speed is not a special speed.
You, me, and a lot of others do misunderstand relativity. In particular your questions makes me think that you see the universe as a“static” space with an absolute “time”, which is not the case. For example 2 simultaneous events can be one after the other if you change the frame of reference.
I found that videos from one minute physics on youtube are useful to better understand relativity. You may start from this one: https://www.youtube.com/watch?v=1rLWVZVWfdY
