I understand this "would" is used as subjunctive mood and "if I was" is omitted before "coming". However, shouldn't the auxiliary verb be followed by the infinitive form of the verb? Why is "would" followed by "if" in this case?

What's "see through" mean? I only know of it as a type of cloth

"See through" something means to recognise that something is not true and not be tricked by it. Ex: I could see through all her lies. Ex: I'm not fooled that easily. I can see through you.

What's "Freee-Inator?"

"Freeze-inator" is not an actual noun, it's just a made-up name that Dr. Doof uses to call his gun. The name has "freeze" in it, so we can assume that it's main function is to freeze things. You'll see this kind of naming pattern a lot with children's toys, like having "splash-inator" as the name of a water gun.

What does "commemorative" mean? In what circumstances could I use it?

"Commemorative" is an adjective! It means that something acts as a memorial or mark of an event or person. It's usually used for big or significant moments in one's life and events that happen. Ex: We got commemorative rings for the ceremony. Ex: The commemorative plaque will arrive at the school tomorrow and placed above the door in honor of our former principal.

What's "hang out" mean? Is it like "play?"

It's not quite the same as "play!" "Play" is considered more childish, whereas "hang out" means to spend time somewhere or with someone for fun or just to relax. Ex: I like to hang out at the library during the weekends. It's very relaxing. Ex: Do you want to hang out with me tonight? We can get dinner somewhere.

I think the word "teacher" is too casual. So, can I say a "mentor" instead of "teacher"?

The use of the term "mentor" depends on the student, of course! However, since we can see that Anne Sullivan's care for Helen Keller extended beyond just teaching and concerned her personal life and well-being as well, "mentor-mentee" can also be used to describe their relationship. Ex: I consider my mother to my mentor. Ex: It was through the help of my mentor that I am so successful today.