I will give you ways to make your writing more simple by writing less (fewer adverbs, shorter complete sentences). Also, when you speak shorter sentences it is easier to do.

One month has passed since I arrived in the  UK

It's been a month since I arrived from Japan.
Studying children's literature is much harder than I expected.
Much more reading, and much more thinking deeply. I need to read more and think more deeply.
But people are nice and I love the nature surrounding me.
I think I can say that  I've started to settle down. and I have become used to living here.

However, I find it really difficult to communicate with other people deeply.
I can understand what my friends are saying, but  I can't say much myself.
I wonder how bilingual people think when they talk. 
Do they think in both languages at the same time?
For me, When I talk to people, I have no time to think in Japanese and translate it to English, so I feel like I am talking without thinking.
I practice by talking to myself in English, but I usually lose track of what I was trying to say.

I have not discovered the most efficient way to become fluent in English, so I need to practice speaking English daily. 
In Japanese, they say "習うより慣れろ, " meaning you learn best by doing.
So I'll try to make opportunities to talk with people in English.

It's difficult for me to keep a conversation going if there is I have nothing in common with the people whom I talk to.
At the moment, I have more international friends than local friends, since we have the same goal: to be a good & clear English speakers. and We also we are facing  face the same challenge: living and studying in the  a foreign country.
I don't think that I can improve my communication skill dramatically, but I'll work steadily.
Starting  I will start by doing my homework with my full effort!

You did a great job sharing your thoughts.

It is not possible to listen to a conversation in English, come up with something to say in Japanese, translate that to English and speak that English translation in the conversation. This is especially difficult to do with Japanese and English because the word order is so different between the two languages. The most efficient/effective way to converse in English is to think in English as you listen, think in English as you form what you want to say, and then say it in English very soon after the speaker stops speaking. Cognitive linguists believe that people who speak different languages actually think differently because they organize and process their thoughts differently according to the particular language they speak. For you to speak English fluently, knowing vocabulary and grammar isn't enough. You must organize and process your thoughts in a way that matches the English language. You must practice thinking whole thoughts (phrases, short sentences) in English.

The Behaviour of My Dog

I have a dog in my house. She is a small Yorkshire terrier. She has been a part of our family for 12 years. At first, I was a little scared to interact with my dog, because it was my first time to have a pet. My dog also seemed nervous when I rubbed /petted/stroked/ (petted is common for American English, stroked is common for British English, however in American English it is common to talk of rubbing a dog's stomach) her. Over the years, I was able to stroke her without the fear of being bitten. I have noticed my dog seemed relaxed in my arms too. I was very happy to find this out. I thought she may have sensed my attitude towards her. How clever a dog is!

You could also say "it was my first time having a pet."

Good job!