Yesterday ,I found Daniel had been very xxx, I realized that Daniel needed to be groomed, so I took him to a Pet Grooming Salon.

There could be several different reasons for your action, so I used a general expression.

I felt very sorry for him, and I picked hime up andto comforted him.

As usual, there are many ways to structure a sentence. This is just another option.

Gradually, he stoped shaking violently.

You don't need to repeat "violently" since we just read it.

So tThe groomer started grooming him.

Even though it may have been a cause and effect (she started because he stopped), I don't think "So" is needed.

The groomer was so skilled,; she groomed Daniel quickly, while calling Daniel'saying his name from time to time.

After about 15 minutes, tThe first step, rinsing finished, took about 15 minutes.

It helps the reader to follow the steps if you use a parallel sentence structure.

Then the second step, blow-dry started, iting, took more than 20 minutes.

Then the third final step, grooming thehis nails and fur, it took about 10 minutes.

The parallel structure helps the reader quickly grasp the order of events and what happened during each step.

In the end, Daniel came back towas once again a tidy, beautiful Pomeranian dog ^_^. ¶
In the end, Daniel came home a tidy, beautiful Pomeranian dog again ^_^.

The second option is a change of meaning, but it uses the word "came" and echoes the beginning of the story where you leave home on this adventure.

Yesterday , I found Daniel had been very xxxwas very shaggy, so I took him to a Pet Grooming Salon.

"had been" feels funny to use here. better not to use a perfect tense. Also, 'shaggy' means his hair was too long, and implies it was messy

I felt very sorry for him and I picked hime up and comforted him.

Might be better to say "I felt very sorry for him, so I picked him up..."

Gradually, he stopped shaking violently.

This isn't wrong but it sounds awkward with the rest of your text. Maybe say "He gradually stopped shaking so violently" or "He gradually stopped shaking." It feels funny to say "shaking violently" again.

The groomer was so skilled, she groomed Daniel quickly while calling Daniel's name from time to time.

"calling Daniel's name from time to time" is an odd thing to say here. I think you mean that she said his name to soothe him, but "calling" is more like yelling. It would feel more natural to say something like this:
"The groomer was so skilled, she quickly groomed Daniel, saying his name to keep him calm."

After about 15 minutes, the first step, which was rinsing, finished.

Then the second step, blow-dry, started, and it took more than 20 minutes.

Then the third step, grooming the nail and fur, it took about 10 minutes.

In the end, Daniel came back toreturned to being a tidy beautiful Pomeranian dog again ^_^.

"Came back" feels more like it is actually going back to a place. "Returned" can be more metaphorical

Groom My Pomeranian

Grooming my Pomeranian

Yesterday ,I found Daniel had been very xxxwas very overgrown, so I took him to a Ppet Ggrooming Ssalon.

overgrown often refers to plants but it can also apply to pets!

I felt very sorry for him and I picked hime up and comforted him.

I picked him up

Gradually, he stoped shaking violently.

He eventually stopped shaking.

So the groomer started grooming him.

Then the groomer started grooming him.

The groomer was so skilled, she groomed Daniel quickly while callsaying Daniel's name from time to time.

After about 15 minutes, the first step rinsing finished.

After about 15 minutes, the first step of rinsing was finished.

Then the second step blow-dry started, it took more than 20 minutes.

The second step was blow-drying, and that took more than 20 minutes.

In the end, Daniel came back to be a tidy and beautiful Pomeranian dog again ^_^.