The Behaviour of My Dog

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

Good job!

The Behaviour of My First Dog

1) No need to say she is female she = female that is obvious.

2) No need to repeat that she is female.

3) Interact with her is better, by this stage of writing you have said she twice, and her once. So the reader most certainly knows by now your dog is female, they have had three hints. (to learn to read or buy glasses or to put them on)

4) Rubbed is fine but it could be misconstrued by some people. Dog lovers tend to rub their dogs stomach, which dogs like. 

"she also seemed nervous when I rubbed her stomach, which I know dogs like" 

5) How clever she is!. Is fine for this informal style of writing it is how people would say it, and it would be understood.

EDIT I forgot to mention (of) owning a pet: - ---- of is optional. 

It indicates a person was most probably born in an English speaking country and is natural to include "of". 

Although often left out in contracted speaking or writing.

https://www.talkenglish.com/grammar/prepositions-of-to-for.aspx

See the owning possessing rule which is the first rule explained in the link.

(Part One)

The Behaviour of My Dog

(Part Two)

Hi Lisa,

I had a male Shetland for thirteen years before he passed away in 2014. I love cats and dogs and their playful innocence! 

Just some comments for your writing: 

(1) Since your second sentence provides more information about your dog (i.e. adjectives such as "small", "she"), I have decided to use them with "Yorkshire terrier" to describe it. Generally, adjectives describing personal opinions are listed before more factual ones. (e.g. "small" describes size which is more subjective; "female" is more factual (less subjective) as a dog is either male or female. 

"I have a small (more subjective opinion) female (factual, less subjective) Yorkshire terrier in my house." --> correct order

"I have a female small Yorkshire terrier in my house." --> incorrect order

(2) If the number is of a small value, we describe it in words.

Example: There are four possible answers for these ten multiple-choice questions. (correct) / There are 4 possible answers for these 10 multiple-choice questions. (the sentence can still be understood, but it distracts the reader with a mixture of words and numbers. 

(3) We use to-infinitives (e.g. to have) to express a future intention. However, you already had a pet as you were patting her. 

(4) We apply some force on a surface using a circular or up-and-down motion when we "rub". For example, we can apply some lotion and rub it gently on the stomach of a baby. I believe you were "patting" your dog by gently and repeatedly touching it with a flat palm.

(5) "Over the years" means something that occurs throughout the years. Since you mentioned that your dog was initially nervous, I changed your phrase to "After a while" to describe a change from the beginning. 

(6) I combined your two sentences since your second sentence (i.e. I was very happy to find this out) is a result of your first sentence (i.e. I have noticed that my dog seemed relaxed in my arms too.). 

Example: I was given a promotion after five years of hard work. I was very happy with my promotion. --> I was given a promotion after five years of hard which I was very happy with.

(7) Our "attitude" describes our mental disposition towards things that happen around us. (e.g. He has a poor attitude towards his work.) I believe you are referring to your display of friendliness to your dog. We can use "affection" for this meaning. 

(8) It is possible to use "How" and "What" to express our surprise at a quality or characteristic. However, the sentence structures differ.

Examples: How intelligent the dog is! / What an intelligent dog it is!

We do not use a noun phrase with "How". (We do not write "How an intelligent the dog is!)

We do not use an adjective immediate after "What intelligent dog it is!)

I hope this helps.