"Three dogs are barking at the store."
Er, shouldn't it be 吠えています instead of ないています? I'm not saying that the given sentence is wrong, because it's not, but I think 吠えています would be a better choice.
REMEMBERING SOME KANJI:
店 - SIDE VIEW OF A STORE (THE TYPE THAT LOOKS LIKE A KIOSK) WITH A PERSON TENDING TO IT
匹 - THE KANJI OF FOUR MINUS THE VERTICAL STROKE ON THE RIGHT (OR: "C THOSE LEGS? LET'S COUNT THEM!" - THE OUTSIDE PORTIONS OF THE KANJI THAT LOOK LIKE A "C" + HUMAN LEGS RADICAL)
犬 - THE KANJI FOR "BIG" PLUS A SMALL DIAGONAL MARK ON THE UPPER RIGHT CORNER
鳥 - COULD BE A PICTOGRAPH OF A BIRD PERCHING INSIDE A CAGE
鳴 - "BIRDS USE THEIR MOUTHS TO CHIRP, YOU DON'T SAY?!"
Yep, it's perfectly fine. I believe that putting the counter first kinda emphasizes the number.
Surely this should accept 1) は instead of が and 2) 店に instead of 店で? Wouldn't 店で imply the dogs are barking IN the store?
Afraid not. If you have the counter first, you need to link it (adjectivally) with the noun it's modifying - hence the の.