Is this sentence written properly?

"Think about it," Link began, "Didn't you realize how much trash there was as the water came?"

No.

You need a period at the end of the sentence -- after "began".

"Well," Poseidon began. "Since they are young, I will give them one more chance."

Is this one right?

Yes.

"Sire! Sire!" The servants busted through the door.

I am suppose to capitalize "The" right?
And are my quotations correct?
Thank you so much ms.sue

Yes, you're right.

However, "busted" is slang and not a legitimate word.

Yes, the sentence is written properly. It is a dialogue where the character Link begins speaking and is followed by a quotation mark. The comma after "Think about it" signifies the continuation of Link's statement. The next part, "Didn't you realize how much trash there was as the water came?", is also enclosed in quotation marks and ends with a question mark since it is a question.