I could see giving this to physics students as a brain-bender. (In fact, I won’t be surprised if it turns out somebody already has.)
In the first scenario, I believe — operating on the remnants of my physics knowledge — that it would accelerate downward. Gravity still acts on the rod; it will move, and the bits of it that pass through the blue portal re-emerge from the orange one with their momentum conserved, so it’s (I think) functionally no different from letting it fall a really long distance. Probably it will achieve terminal velocity at some point.
In the second scenario, I think the rod would accelerate the other way? But I’m not sure. The falling orange portal would push some of the rod back out the blue portal, which pushes more into the orange portal, and you’ve basically got the same situation as in #1, except in the other direction. But the part I can’t figure out is what happens when the orange portal comes to rest atop the blue one. (Or even not directly atop it — you could stop any distance away that is less than the length of the rod.) Does the rod bend? There’s no longer enough room for all its length between the portals, so I feel like it must, but I’m not sure how the force for that works out. (And actually, if the rod is allowed to move as in scenario #1, then I think you get this problem right away. Because then the rod is trying to come and go from both portals at once.)
In the third scenario, I think that if the portals are shot as depicted in the diagram, you’ve made a weak projectile. Move the orange portal, and now the rod falls through the floor and out the wall. (If you’ve let it build up momentum via the first scenario, then maybe it’s not so weak.) But that assumption depends on what I think is an as-yet unanswered question in the games, namely, what happens if a portal goes away while something more solid than a beam of light is athwart the boundary. I’m presuming it severs the object in question, so that you’ve basically made an ordinary piece of pipe with a solder in the middle, which then falls through the blue portal. I’m not sure we ever saw that issue in action during the game, though, at least not as a puzzle. (Probably people have left turrets or cubes balanced on the portal boundary and then shot a new one; my guess is they fell to whichever side had the majority of their mass. But that may just be a coding default, rather than a conscious choice on the part of the designers to say that portals can’t slice objects in half.)
It’s been years since I thought about this stuff, though. Tell me, O internets: where have I got it wrong?