Vedo un olivo in giardino.

You could, but Italians rarely use the present continuous with perception verbs.

• Vedo un olivo in giardino sounds natural.
• Sto vedendo un olivo in giardino is grammatically correct but unusual; Italians prefer the simple present for “I see.”

Italian has two masculine singular indefinite articles:

• un before vowels and most consonants (e.g. un olivo, un libro)
• uno before s+consonant, z, gn, ps, x, y (e.g. uno studente, uno zio)
  Since olivo begins with a vowel, you use un.

In Italian, certain location phrases drop the article after in as an idiom:

• in giardino (in the garden)
• in ufficio (at the office)
• in chiesa (in church)
  You can add the article for specificity, but the default is to leave it out.

Yes.

• In giardino is the general, idiomatic form for “in the garden (of one’s home).”
• Nel giardino (in + il → nel) emphasizes a particular garden or is followed by more detail (e.g., nel giardino di nonna).

Italian assigns prepositions to places by convention:

• in with open spaces or institutions (in giardino, in biblioteca, in aeroporto)
• a with towns, some rooms or activities (a casa, a scuola, a tavola)
  These pairings must be learned individually.

Yes. Italian word order is flexible.

• In giardino vedo un olivo places emphasis on where you see it.
• The neutral order is Vedo un olivo in giardino, ending with the location.

Subject pronouns are optional in Italian because the verb ending indicates the subject.

• (Io) vedo un olivo in giardino is correct but redundant.
• You include io only for emphasis: Io vedo un olivo, meaning “It’s me who sees an olive tree.”