Eu estudo música popular com o Pedro.

In Portuguese, when naming a field of study or talking about something in general, you often drop the article.

• Eu estudo música popular = “I study popular music” (as a genre/disciple in general).
  If you include the article (Eu estudo a música popular), you’re usually referring to a more specific set or body of works within that genre.

In European Portuguese it’s very common to put the definite article before personal names:

• o Pedro, a Maria, os João e Luís.
  This is a regional norm that adds a familiar or colloquial tone. In Brazilian Portuguese or in very formal registers you might skip it (com Pedro).

Portuguese adjectives usually follow the noun:

• casa grande, livro interessante, música popular.
  Putting an adjective before the noun can add emphasis or literary flair, but the neutral order is noun + adjective.

Yes, but note regional differences:

• In Brazilian Portuguese you’d often say estou estudando.
• In European Portuguese the progressive is usually estou a estudar.
  You can also use the simple present estudo, which in Portuguese covers both habitual and ongoing actions.

No. Verbs in Portuguese are conjugated for person, so estudo already tells you it’s first-person singular. You can simply say:

• Estudo música popular com o Pedro.
  You include eu only for emphasis or clarity.

Here com means “with,” indicating accompaniment:

• com o Pedro = “with Pedro.”
  Portuguese com can also be used for instruments (e.g. toco guitarra com uma palheta) or ingredients, tools, etc., but in this sentence it’s the straightforward “together with someone.”