If an Immovable Rod is made to stay still, and a Bag of Holding is placed around it such that the Rod is closed entirely in the nondimensional space, does the Bag hang on the Rod, or does the Rod move around with the Bag (and if so, does it remain floating motionless in the Bag)?
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Nice theoretical question. :) And I partially disagree with the other two (?) answers (that I've seen so far.)
I'd say you could place the Bag of Holding around the Immovable Rod, or push the Rod into the Bag, though with a very, very serious effort -- a DC 30 Strength check -- only. Why? Because
...and the rod does not move from where it is, unless you apply serious force (a DC30 STR check, as per the rules.) Pulling the bag over the rod or pushing the Rod into the Bag would cause the Rod to move, from the original dimension to the bag's dimension. It would be like trying to touch two similar magnetic poles to each other - which can be done with great enough force. So, you could pull the Bag over the Rod, etc... with a DC30 STR check.
If you activate the Rod within the bag, its position will get fixed within the Bag's "nondimensional" space (whatever that means) only, so you'll be able to move the Bag anywhere on the outside, with the Rod in a fixed position in it.
In case you activate the Rod when it's only partially covered by the Bag, moving the Bag off the Rod, or moving the Rod along with the Bag would be a DC30 STR check, since the Rod would have a fixed position both within and without the bag. (Well, depending on your view of physics and magic, you could lower the DC in such a case to your heart's content.)
Since the bag of holding specifies that it utilizes its own, non-dimensional space, the immovable rod may not move within that separate dimension, but the dimension itself is moved along with the bag of holding.
For example, consider anchoring a boat in a bathtub. Your boat is anchored, but you can continue to move the tub around.
Clarification: To get an immovable rod to remain stationary and inside of a bag of holding, you may need to reach into the bag with the rod off, and then activate it. See also: OpaCitiZen's answer
I believe that the motionlesness applies to the plane (or dimension) in which it lies. Since a bag of holding is an extradimensional space, in effect a different plane, although a small one, the rod will remain immobile inside the plane while the entrance to the plane, the bag, may move freely.
Activating the rod in the air and fixing it in space in whichever world you are in SHOULD affix it to that space IN THAT WORLD (i.e. to the planet or body you are standing on, otherwise it could rip the planet as it moved through it while orbiting a sun). Placing the bag over it causes the extra-dimensional space to move around the bag, not the rod to move in the space. I think the power of the user who created each object MAY come into play, but, DM's call here, it will either:
1) STILL be fixed in the original spot, and the bag can hang on it, because, it never truly left the 1st plane of existence. (if the power of the bag's maker is lower than the power of the maker of the rod) or
2) The bag will be able to pull the rod away, because it is tricked into thinking it is still in the same space. Upon removal from the bag, it will INSTANTLY resume it's position in the ORIGINAL world/plane. -This would mean either:
2A) vanishing and instantly reappearing in the original spot, OR,
2B) it instantly MOVES to it's original spot as the elastic connection between the black-hole type vortex of the bag releases the warp in space between the two dimensions, leaving a rod-shaped hole in EVERYTHING in it's path (Warner brothers-style, i.e. coyote-shaped holes) be it a building, a mountain, a wagon, or even the horse! Unmovable becomes unstoppable, but taking no time in the transition.
Given a few seconds of prep, being chased down a narrow canyon could be fun if you think about it! Traveling far enough to pass the horizon and go over the curve of whatever planet you might be on could form a new creek, or going farther, cause a very small volcano to form on both ends.