What would moving in the 4th dimension look like in 3d? I've been reading "Shape of Space" and watching videos from the videogame Miegakure. Both talk about >3 dimensional space.
I'm not sure if Miegakure's interpretation is accurate and it's limited to youtube only https://www.youtube.com/watch?v=XfiFBsKi7go.
Say I have a cup of water sitting on a table. Is it safe to say that the cup, the table and me are all in the same position in the 4th dimension?
Secondly, if I began moving the cup into the 4th dimension, what would I see? Would it start to fall through the table? Would my hand be able to go through it?
Or would it disappear instantly when it moved any amount in the 4th dimension?
Similarly, when it came back.. if I had my hand in the way, would I feel it push against my hand? Would it push my hand out of the way? Could it expand within my hand from the 4th dimension or would it simply not reach because I was in the way?
 A: Hard to imagine. However we can imagine living as inhabitants of a 2D world, which is embedded in a 3D world. That 2D world could be the screen you are looking at.

Say I have a cup of water sitting on a table. Is it safe to say that
  the cup, the table and me are all in the same position in the 4th
  dimension?

Well in my reduced example, you would see an image of a cup, table and yourself on the screen. Now all the objects have 3D coordinates $(x, y, z)$,
where $x$ and $y$ might be horizontal and vertical positions on the screen and $z$ is orthogonal to the screen, e.g. it could be the distance from the screen. All the mentioned objects share the same $z$-coordinate $z = 0$, as they stay on the screen.

Secondly, if I began moving the cup into the 4th dimension, what would
  I see? Would it start to fall through the table? Would my hand be able
  to go through it?

In the reduced example:
The question is would all or only part of the cup move into the third dimension? 

The cup is flat (*). You could move that flat cup all to $z = 1$ then it would hover in front of the screen, above it, parallel to it.
You, in a similar situation like ours, would only be able to perceive the 2D world, what is on the screen. The cup would have vanished from your view.

If you just rotate the cup into the 3D world you would have a part above the screen, a part below the screen and the rotational axis part stays with $z=0$. The 2D cup would shrink to a 1D line in your view.
Update: (*) Is it really flat? What if it wasn't flat the whole time? But for some reason we can only see and interact with that 2D-slice which is part of our 2D world. So it might change shape too.

Or would it disappear instantly when it moved any amount in the 4th
  dimension?

Only if all of the points that it is made up, get $z \ne 0$.

Similarly, when it came back.. if I had my hand in the way, would I
  feel it push against my hand? Would it push my hand out of the way?
  Could it expand within my hand from the 4th dimension or would it
  simply not reach because I was in the way?

No idea how physical forces would act. It might squeeze you out in the third dimension.
A: Thought experiments like these can always be made easier by trying to think first about how one might pass from a $2$-D environment to a $3$-D one. So, think about a table, a chair, a cup of water on the table, and Mario together in a $2$-D room with mushrooms. If the cup of water is pushed out to the third dimension, what happens? Well, disregarding gravity, it just gets pushed out. So it doesn't "fall through" the table which is lying in what is now a different $2$-D plane. So, prior to the cup moving out to the third dimension, all the objects lived in the same plane, but now they live in different planes inside of a three-dimensional space.
Similarly, if we now visualize you in a $3$-D room with a table and a cup of water, then yes, your intuition about the cup "disappearing" in an instant into the $4$th dimension applies. The cup now resides in another $3$-dimensional subspace of $4$-D space.
