Is it possible for the Player Character to become immune to radiation poisoning?

I haven't progressed that far into the story, so my answer won't be much of a spoiler, but as far as SPECIAL and perks are concerned, it is impossible to be entirely immune to radiation.

Using a console command, I got my character to a high enough level to purchase all perks. Although my character was then able to swim without receiving any rads, and regenerate both health and rads, she was sadly far from immune to the effects of the Glowing Sea, and had to consume a number of Rad-Aways to avoid dying.

I don't know of anything that you can get which is connected to the Children of Atom, however it's possible to gain access to a consumable that will remove 10 rads a second for 1h. You will in practice be able to keep this buff going forever because of the long duration. This will basically make you immune to rads, but technically it's not true immunity.

Spoiler below on where to get it:

Complete the quest "The Secret of Cabot House" by siding with Lorenzo at the end instead of killing him. This will grant you access to the consumable "Mysterious Serum". He will give you a new one every time you spent the one you got from him last, so you have an infinite supply as long as you're willing to take the trip to his house every time.

Console commands:

player.modav 2ea 10000+ (radiation exposure resist)
player.modav 2e9 10000+ (rads from food)

Yes and no.

There is almost certainly a console command that will add whatever perk or attribute Super Mutants and (presumably) robots have that renders them immune to radiation to the player, but I can't find the ID online. It should be easy to find once the GECK is released. Failing a direct console command, it will certainly be moddable. Immunity to radiation is an existing mechanic, so it should be trivial to make available to the player via mods.

Failing that, there are ways to become effectively immune to radiation (power armor w/ lead shielding and mysterious serum work, as do other gear and perk combinations), but due to the way rad resistance works, you can't become absolutely immune to radiation.

Rad resistance appears reduce incoming radiation by a percentage, rather than flat amount, as in previous games. 100 Rad resistance is a ~50% reduction, but 200 isn't significantly higher, and you still take rads at 1000. The equation has yet to be determined to the best of my knowledge, but it seems likely that there is no number that will reduce radiation actually taken to true 0. Some unbelievably tiny number that we would round to zero in the real world, perhaps, but not zero. Any sufficiently high number is probably well beyond the maximum obtainable Rad Resistance available in the base game. And if we are going to use mods or console commands to increase our Rad Resistance artificially, we might as well just give ourselves immunity anyway.

Yes (assuming you have the Far Harbor DLC), but it comes at a cost:

The Robes of Atom's Devoted give resistance increasingly as the protagonist becomes more irradiated. This means that as their maximum health decreases, their radiation resistance can go up to 2500 and above. The robes can be lined with Ballistic Weave.
Additionally, due to a bug, the maximum resistance can be maintained even as the protagonist loses his or her rads once they have brought it up to 900. See the link for more details.

Mod a Power armor suit with lead plating, way higher rad resistance than the HazMat suit. 1600 I believe