Plenty of people don't even put their nic in the freezer at all, opting to keep it in a cool dark place like a basement or cabinet in the garage. It's a safer bet to have it stored in the freezer but taking it out for a day or even a month doesn't cause it to degrade in any significant way. Keeping it on ice may be better than not but it's not going to be noticeable for that amount of travel time.

Ship it to your destination, pick it up, put it back in the freezer. No need to over complicate things :)

You actually can't check that at all. It's hazmat and you can't fly with that.

I'd ship it to yourself if i was you.

im sure it will be fine at room temp

if your worried, the most I would do is wrap it in a tea towel with an ice brick, depending on the brick it may still be frozen but even the shittest type will still be cold

Are you moving? What's the need for bringing an entire 4L of Nic with you? Did you not decent into smaller bottles?