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Four AA batteries.
I don't think he have calculated practicality or energy use yet, but for a theoretical/philosophical exploration of Portals, you'll need to go to: https://www.youtube.com/@optozorax_en/videos
If portals were real there could be tiny portals in the portal gun to run power or vent heat so it could be as light as you want it to be.
Yeah quantum energy teleportation is a scientifically confirmed possibility.
To answer that, one would first need to answer how much energy is required to make a portal. That is a tricky question to answer, because that tech doesn't exist.
Since you said 'realistically' let's assume it's using wormholes (which are in fact "possible" arrangements of spacetime in General Relativity). Note that actual wormholes would not be flat, they would be spheres, but that's besides the point.
Best number I could find for the energy needed to hold a person-sized hole open is from Here.
"negative mass-energies on the scale of Jupiter are required (-1.898 x 10^27 kg)"
Note that we don't have a method to produce Any negative mass (antimatter is still positive mass-energy).
There are things like the Casimir effect but you can't take that negative energy and move it elsewhere or concentrate it as far as I understand.
So the answer to your first question already puts this Way out of the realm of having meaningful answers to the others.
Erm, it's not really possible to answer this question "Realistically"
Since portals aren't real, maybe we can start from what we see in game: Chell can hold it for long periods of time without problem, so that puts an upper weight barrier on it. Aperture must've figured out a way to fit portal technology into it, or found a way to stash its mass away. Or maybe it's heavier than average, but Chell is drugged to be able to hold it.
There's no power cord in-game, so its energy source must be self-contained, because I don't think it can get enough power by absorbing things in the air.
It's important to note that Chell has reinforced legs. That is important for the whole weight distribution thing. She can probably lift heavier things or at least become less fatigued carrying something for long periods of time.
She doesn't have reinforced arms and is able to aim it precisely over those same periods. I do like the idea that heavy components e.g. massive amount of power are connected to it remotely via the portal tech itself, but what we see of it on this side that's in Chell's hands is definitely lightweight.
I think you meant Aperture
thanks! I've fixed it.
Regarding a power source, we only use it within Aperture, so it might be powered through some kind of induction charging in the walls.
Or the power source is somewhere else and the gun gets its power with a cable running through a tiny portal.