Yes, storing your e-liquid in the fridge is a great way to make it stay fresh longer... but only if you don't plan on vaping that bottle for a couple of years. I do keep my mixing supplies in the fridge (and freezer) to prolong their lives, but we're talking GALLONS of supplies, not 30ml bottles. now, if I mix a batch of 1-liter of e-liquid, and find I don't like it after I've vaped it for a couple months... I may store the remaining dozen 60ml bottles in the fridge so they won't go bad until I decide they're worth vaping again... but even when I mix up my ADV (yes, usually in 600-900ml batches), I don't store them in the fridge... they're on a shelf in my office, which is cool and dark... and I'll normally go though those 10-15 60ml bottles in a few months... so they don't go bad while waiting to be vaped.
As for batteries... the old-school dry-cells made prior to the 1970's did "last longer" if they were kept cool, like in the fridge, but today's more advanced Alkaline batteries aren't helped in the least (usually harmed as @Bebop stated above) by putting them in the fridge. As long as they are kept cool... like normal room-temps... they'll be just fine in storage for more than 5 years.
On a side-note... I bought a box-lot of vintage cameras a few years ago, and the Brownie camera flash had EveryReady batteries in it... from the early '50s... and they STILL WORKED to power the flash bulbs, and hadn't leaked in 60+ years of storage in someone's basement!