You need to realize that a verb can be either
- finite (= matches the number of the subject of the sentence) or
- nonfinite (= no inflection), which are infinitive, participles and gerunds. We are dealing with infinitives in your examples.
For an independent clause (AKA "sentence") you need at least one finite verb and unless you are making a "list" ("She laughs and sings and dances.") you use only one finite verb. In your example sentences you use four finite verbs, do, play, like and have.
Very simple sentences use only one verb, which by definition must be finite.:
He has the bottle.
They have the bottle
For questions or special emphasis you use an auxiliary verb (-> finite) together with a verb in the infinitive:
He does play cricket.
Do they like cricket?
So yes, in these cases "do" becomes "does" for third person singular because it is finite.