的 generally expresses possession or modification and 得 is used after a verb or an adjective to express possibility or capability.
Here are some literal translations that should help to differentiate between the two examples that you listed:
[The Chinese that he speaks][(is) fluent].
In general, that which comes before the 的 modifies what comes after it:
[The clothes that he wears][(is) fashionable].
The 的 ends up forming a complex subject:
The Chinese (that he speaks)
“他说” modifies "汉语".
Which Chinese? The Chinese that he speaks.
The clothes (that he wears)
"他穿" modifies "衣服".
Which clothes? The clothes that he wears.
The (red) book
"红色" modifies "书".
Which book? The book that is red.
得 is generally used in the following pattern to describe the degree to which something is done:
With the dialogue that you provided, the "的" should actually be "得". It seems to just be a typo.
Lily: "Am I small?"
Monster 说：“小？ 你？ 你小的不能再小了。你非常小。”
Monster: "Small? You? You're so small that you can't get any smaller. You're very small."
With the third example, 得 should not be used. 的 is correct here, but it can be confusing because there is an implied/omitted noun. The thing being modified isn't said explicitly and all that is said is the modifier.
[I][don't know][(that which) I did][is/is not right].
I don't know if what I did was right.
Here is another example dialogue:
A: I want to buy books.
B: New (books) or used (books)?
A: New (books).
Notice how "书 / books" is not repeated. Once the object is clearly in context, both speakers drop it for convenience and brevity. Natural human speech tends to be like this. It would seem quite odd to say everything explicitly since once something is in context, repeating it can seem unnecessarily redundant.