Typically, only the first verb in each clause can be finite:
I play.
I played.
She plays.
She played.
As you can see, this verb changes form. It shows tense (present or past) and agrees with the subject (adding -s if the subject is third person singular present). This must be our finite verb.
Our example is a declarative clause. To make it into a question, we should turn it into an interrogative clause. And for this sort of sentence, we do that by swapping the subject and auxiliary verb, like this:
She was safe. → Was she safe?
This is called Subject-Auxiliary Inversion (SAI). We use SAI to mark these sentences as questions. And in this case, SAI is possible because we have the auxiliary be.
But what if our example doesn't have an auxiliary verb? Let's look at our example from earlier:
I play.
I played.
She plays.
She played.
We can't swap the subject and auxiliary verb if we don't have one! Let's solve that problem by adding the meaningless ("dummy") auxiliary do:
I do play.
I did play.
She does play.
She did play.
Now we've got two verbs! And only the first can be finite. In this case, that means do! And as you can see, do now changes form, but play doesn't! That's because play is no longer finite.
Now we can make it into a question. So, let's do Subject-Auxiliary Inversion again, moving the auxiliary before the subject I:
Do I play?
Did I play?
Does she play?
Did she play?
These questions are formed correctly. But what about your example?
*Does she plays?
It doesn't work! Why not? Because we'd have two finite verbs.
In this answer, the * symbol marks a sentence as ungrammatical, and bold in examples marks a verb as finite.
An open question is one that has many possible answers: subject or object questions are examples of this, as the there are multiple possibilities.
A closed question requires a yes or no answer.
Closed questions in the past simple are usually asked using **did*.
Did you break it?
I hadn't thought about it before, but I think that you have identified a pattern, in that we also use did for object questions, presumably so that the object pronoun can be moved to the start of the sentence.
Best Answer
I (1st person singular) do like words.
A kitten (3rd person singular) does like words.
If you want to stress the verb in the second sentence you'll probably want to use an adverb. "A kitten truly does like words." The word do in your first sentence is part of the verb, not a modifier. I can see how you might be confused if you are learning English the way it is generally taught nowadays, i.e. without a formal study of English grammar.