Mi piace ascoltare musica dal vivo in giardino.

When you talk about an activity or something in a general, non-specific sense, Italian often omits the article.
• ascoltare musica = “to listen to music” (in general)
If you refer to specific music, you could say ascoltare la musica or add a qualifier like ascoltare la musica classica.

dal vivo literally comes from da + il + vivo. It’s an idiomatic adverbial phrase meaning “live” as in “performed in front of an audience.”
• da = “from”
• il contracts with da to become dal
• vivo here isn’t agreeing with musica; it’s part of the fixed expression dal vivo (“live”).

For locations like giardino, Italian normally uses in without an article to express “in that space.”
• in giardino = “in the garden” (generic location)
Using al giardino (from a + il) would sound like “to the garden” or refer to a very specific garden, which isn’t needed when simply stating where you listen.

The subject of piace is the infinitive clause ascoltare musica dal vivo in giardino—that entire action is what “pleases.”
• mi is an indirect object pronoun meaning “to me.”
• piace agrees with the subject (the listening activity), not with mi.

Yes. Because ascoltare musica dal vivo in giardino is the subject of piace, you can front it for emphasis:
Ascoltare musica dal vivo in giardino mi piace.
The verb piace still stays in third-person singular.