My childrens

(Children - это уже множественное число от слова child, к нему не нужно добавлять окончание -s)

Me and my family watched the animation film "Madagaskar".

(show - показывать, look и watch - смотреть. Ваша семья показывала или смотрела фильм? Также нужно не забывать ставить артикли перед перечисляемыми существительными)

This movie liked my childrens very much, especially my daughter.

My children, especially my daughter really liked the movie.

(У вас получается, что фильму понравились ваши дети. Скорее всего наоборот, детям понравился фильм?)

The children laughed very loudly. It was very funny.

My daughter's name is Anna. We call her Anya. She is seven years old. She goes to a school in first grade. Anya goes to a swimming pool too. She likes to swim very much.
My son's name is Artyom. He only two years old. He learns to speak. In the new year he will go to a nursery school. He has white hair and gray eyes. Artyom can run very fast and often falls.

My childrens

My family and I watched the animation film "Madagaskar". This movie like my childrenThe children liked it very much, especially my daughter. The children laughed very loudly. It were very funny.
My daughter's name is Anna. We call her Anya. She's seven years old. She goes to a school  and she is in (the) first grade. Anya goes to a swimming pool too. She likes swimming very much.
My son's name is Artem. He's alost two years old. He is learning to speak. Next year, he will go to a kinder garden. He has blondy hair and gray eyes. Artem can run very fast and often falls.

My family and I=Subject

To watch=to look at somebody or something for a time

To show= to let somebody see something