I am confused on the use of good, well; real, really; bad, badly.

Cats can see good/well in the dark.
I think well because it describes how they see(an adverb)

After the storm the pier was damaged bad/badly.
I think badly because I tells how it was damaged ( an adverb)

Yay! Both are right.

I agree with both your choices.

How about these?

I was real/really pleased to get your invitation.

Really because it describes how pleased I was?

Sylvia looks good/well in her mermaid costume.

Well because it describes how she looks.

The first is right.

The second should be "good" because "looks" is a linking verb in this sentence. "Good" is a predicate adjective describing "Sylvia."

The first one is correct.

The second one isn't.

If you say "she looks well" then you mean she can see just fine.
If you say "she looks good" then you mean she is not ugly!!

Same with the verb "smell" --
If you say "the dog smells badly" you mean his sense of smell is off.
If you say "the dog smells bad" you mean he stinks and needs a bath.

One more,

In the 1830s the Cherokees were treated very bad/badly.

Badly describing how they were treated?

Right.