Yeah, I keep them on a shelf in my basement or the coolest room of my house. Haven't had an issue with my PG or VG.
Yes, your freezer cannot freeze PG or VG, but it gets cold enough that they will get quite thick and hopefully slow down the nicotine & oxygen reactions.
I use an oral dosing syringe from Walgreens that comes with a stopper that fits into these 120ml bottles I have very snugly. (They also come with a brush for cleaning that works great for all my syringes.) So I'm not dripping out of the bottles. They sit there with their stopper and the syringe is in the stopper and I just flip, draw, un-flip, pull syringe off and dispense into my mixing bottle. (All the bottles are recapped, with screw caps, when I pull the stopper out and put things up.)
Once you shake that bottle, you will be inducing more air. If you have a higher VG content you'll be able to see the air bubbles from shaking. I'm not really worried about the air in the bottle for the short time the bottle is in use, just the outside air over a long period of time.
I have one recipe that actually gets better if it is left open for awhile, just one, and I hear plenty of folks who confuse steeping with airing-out that say their stuff gets better with exposure to air. So I'm not a big fan of air, but remain open minded.