1. both expressions are accepted, although also the first one is correct 2. constraints due to the music and rhythm (metric) ? 3. He is free to write as he likes, and he liked this way that gave the Author a nice trick to avoid a word repetition

Dear Dima, the use of don't instead of doesn't for the 3rd person to negate the present tense is very common in the south. When musicians like Led Zepplin and the Rolling Stones and Sting set their eyes on southern music and jazz styles influenced by the likes of Muddy Waters, Robert Johnson, etc., they also started incorporating the southern colloquial in their singing styles. Even the Beatles sang "she's got a ticket to ride and she don't care (my baby don't care)."

Sting is a poet. So we are not looking for his choices of English to be correct. We must look at the music and the word relations. If we looked at the a music, we may find that the third if a second saved with by saying don't rather than doesn't helps if fit into the time of the stanza. If you look at the word doesn't we see it has an s where as don't has none. The sentence also ends with s. Hearing the repeated s there may be a sound he wanted repeated. He is not looking for consistence like a rhyming poet; he is looking for a uniquely told story. I do not presume to be able to see all the intricacies of stings writing, but this is a start for you as you try to unlock his art.