[OzukakiBurasuki]

Japanese people always leave out so many words when we speak Japanese, that I try to be careful not to leave words out in English, but I don’t seem to know how to write English naturally.

It's okay. There are many instances in which we will leave out certain words, too. It usually just is because of slang.

“The company gave notice to me every year” Is this right?

Yes, that would work. The only problem is the context you are using it in. For instance, if you were still going to pay every year, you would of used "gives" instead of "gave." Since you thought it would be fine to stop paying, it is in context to mark that you won't be paying anymore, it would be "gave" instead. If this might be a little confusing, how about this? If you are still going to pay, you use "gives." If you aren't going to pay after this year, you use "gave."

Sorry, I don’t understand.
“I would need a grave in a yard if I lived in the countryside.” sounds different from what I wanted to mean.
It sounds like that “I live in the city, so I don’t need a grave.”
Should I have written “I would be able to have a grave in a yard.”?

I'm sorry, but I must of read your post in the wrong context. Sorry for confusing you. It is confusing the way you worded that section of your post.

"If I lived in country side and I had enough land to bury them, I would have a grave in a yard, but here in Kawasaki, people don’t have such space, so we call a pet cemetery company when our pets die."

Try wording it like this:

"I would of buried them in my yard if I lived in the countryside because there would be a lot more space. Since I live in Kawasaki, I don't have enough space, so I must call I pet cemetery company instead to bury my pets."

This would of been way more useful to understand. At least you got the point out though, but it was confusing.

OK. Short sentences are better.
I used to be told to connect sentences in English conversation classes. I thought that sentence was not long, but it was too long. OK.

Well, there's nothing wrong with longer sentences, but you could accidentally make grammar mistakes. Because of this risk, it would be more cautious to keep your sentences small with maybe only two main clauses; connected by a comma.

I didn’t know you prefer apostrophes. Thanks.

Yeah, it is great to use apostrophes more than using "of" because it would be much faster to speak and comprehend.

I don’t understand “Many American readers would probably not expect this and would write the same thing,” This sentence wasn’t supposed to be written, and you would write the same thing?????

I was just mentioning that it would be a more common mistake in English. Since "of course" can be taken out of the sentence as it's own clause without disturbing the entire sentence, it would need to be within two commas. Don't worry, it isn't something to get confused over. ^_^

Thanks a lot, OzukakiBurasuki.

Thanks for saying thanks.