Once upon a time, there wereas a gentle old married couple, the gentleman( Ojiisan) and lady(Obaasan).

They way you wrote it was a bit awkward, and I think "was" should be used because they are mentioned as one unit, a couple.

One day, they saw a puppy, which wereas soaked with rain in front of their house.

You should use "was" because it refers to the puppy only.

They decided to live with athe puppy who wasm they named "Shiro" by them."

"whom" is used because it refers to the object of the sentence, which is the "puppy."

After Shiro grew up, he took Ojiisan somewhere and barked " Dig here, dig here"."

Commas and periods go inside quotation marks, while colons and semicolons do not.

Ojiisan dug in the ground, and unearthed a hoard of gold coins was unearthed.

Your sentence is a run-on sentence called a comma-splice because two independent clauses are connected. That is why I used "and" to connect them. Then I placed unearthed right after "and" to keep it in active voice and not in passive voice, since the first part of the sentence was in active voice.

Their neighbors, a mean Ojiisan and Obaasan, looked at their gold coins and borrowed Shiro to use for finding gold coins for themselves.

Unfortunately, Shiro didn't bark, and just stopped.

You should separate two verbs with a conjunction and not with a comma and adverb.

Then, athe mean Ojiisan misunderstood about Shiro's action and dug there, but he only found a ton's of garbage.

I would use "the" because you already specified who the character is earlier in your story.

AThe mean oOjiisann was mad at Shiro and beat him.

AThe gentle oOjiisan burst into tears in front of Shiro's grave and called "Shiro, Shiro"."

Then, suddenly a branch, which was pierced in the grave, grew rapidly and became a log.

AThe gentle Ojiisan made a mortar used from that log.

He and his wife pounded steamed rice in cake using athat mortar.

I would use "that" because you are talking something very specific and used in the previous sentence.

Unbelievably, what they saw was full of .......

I don't know how you plan to end that sentence, but the beginning seems very awkward. I changed it a little bit, but it may still be inappropriate depending on how you finish the sentence.