The point is why ?

1) No. "What makes the earth rotate?" is correct.

2) The second sentence is incorrect. The infinitive is understood, but not expressed.

3) "What makes the earth rotates" is wrong, because "rotate" is the understood infinitive, which is not conjugated. Yes, "earth" is third person singular, but it goes with the verb "makes", which matches "earth" in person and number. 

Maybe this should be moved to Words and Wordplay?

1) "to rotate" is grammatically correct, but it's customary to omit the "to" where I live, when the verb is "make."  However, we would say "What causes the Earth TO rotate?"

2) "Earth" is the object of the sentence.  "rotate" is used in the simple tense.  In the two sentences

This is what makes a bird fly.
This is what makes birds fly.

The verb is "makes," and applies to the subject "what."  The simple tense is "fly," not "flies" in both cases, does not have a plural.  Takes getting used to, especially to people whose first language didn't have plurals.  After listening to a lot of English, one way will sound right and familiar, though.

In 1), there is no need for "to" and many would call it wrong, as well as unnecessary. For 2), the answer about 1) is self explanatory. In 3), it should be "rotate" because, while it is true the earth rotates, once "makes" (whatever, like "the Earth" is inserted, the verb form (of to rotate) switches from "rotates" to "rotate."

go look up the use of a blind or naked infinitive with modal verbs (when the infinitive is said without the "to" part, it is called "naked" or "blind").  For example, the sentence "You must go now" employs a blind infinitive (the "go" is an infinitive, but because it follows the modal verb "Must", it is stated as a blind infinitive). "make" as in force/oblige/impose works as a modal verb.

Make me run, if you can.

"What makes the Earth (to) rotate?" [to rotate = verb]
"What makes the Earth warm?"        [warm = adjective]

"Make" can take the infinitive form of a verb as a kind of verb-adjective.  
"Makes ... to rotate" and not "makes ... rotates".

However, the complication is that in modern English we tend to leave out "to" when using "make", so it becomes "makes ... rotate". Using "to" is not wrong but it sounds like 100 year old speech.

As another answer said, we still say "causes ... to rotate". So as usual, the most common form ("make") is the one that changes first over time.

Verbs like 'make + object + verb' go like this:

My dog makes me laugh.

The teacher makes us work hard.
My mother makes me help with housework.
I don't know what makes you sad.I don't know what makes the Earth rotate.

1 "What makes the Earth rotate? "  is correct

> "What makes the Earth rotates" wrong ?

Because a phrase can have only one "finite form", and in this one the subject is *what* . "The earth" here is the object

looking at common usage, you see "What makes the Earth rotate", occasionally "What makes the Earth to rotate" (more likely "what causes the Earth to rotate"), as both are correct, but the first is more used.

"What makes the Earth rotates" is wrong, there is some subtle rule of grammar here.
even 'what things makes the Earth rotates" is wrong.