Compound verb in: He likes to sing and play

There is no compound verb in sentence 1.
In this sentence, the subject is he, the verb is likes and the (direct) object is (to) sing and play.
Infinitives may, of course, be used as nouns, and are so used here.
The other two sentences have the same verb (likes) repeated. I think that awkward.

I prefer sentence 1 with only one stated verb . That is all that is needed.

You can coordinate two or more words or phrases of the same grammatical category to create a new word or phrase of that same category. Your example 1 coordinates two verbs to create a verb "sing and play". Your example 2 coordinates two verb phrases to create a verb phrase "likes to sing and likes to play". Your example 3 coordinates two sentences to create a sentence "He likes to sing, and he likes to play."

I used the term "coordinate" instead of "compound" to avoid confusion with the term "compound verb" used to refer to a single word verb made by combining two words, like "stirfry" (see Compound verb).

In using the term compound, it may help to keep the distinction between the lexical classification of a word and its role in a sentence. The OP's example has a compound direct object

to sing and [to] play

A direct object has a function in a sentence (namely to take the action of the main verb), and it's composed of two of infinitives (which are words of the lexical class verb). An alternate expression could contain a compound main verb

He plays and sings

Compound verbs, on the other hand, are conjoined verbs that act together as a single verb with its own independent sense. These are rare in English. One response here has suggested

bow and scrape

but this is really a set phrase of two separate verbs, with scrape meaning to bow awkwardly by drawing the foot back across the ground.

Another suggestion is stirfry, but this really means stirred frying, with the first verb acting as a modifier to the first.

About the only pure compound verb I can find is go and do, meaning to deliberately perform an action, usually one frowned on:

He called the boss a jackass? Did he really go and do that?

Note that this verb is an idiomatic unit, different from

Did he really go to church and do an act of contrition?

Also note that both parts of the compound verb take tense:

Yes, he called the boss a jackass. He really went and did it.