In Korean, although possessive pronouns (I, you, we, he, she ...) and adjectives (my, your, his, her, our...) do exist, they are often omitted from sentences when the meaning seems obvious from context.
In an 1:1 dialogue, it is most likely that 1st and 2nd person possessive pronouns/adjectives are dropped. Names and titles are preferred to refer to 3rd party.
Even the possessive particle 의 is seldom used unless talking about some big entities (club, company, country ...).
Having said all that, it seems harsh to fault someone for using possessives in sentences because after all syntactically, it is not wrong to do so.
In Korean speech language, more often than not, present simple tense is used to describe current situation even when this maybe represented in present progressive in English.
The -고 있어요 progressive form is only used for emphasis when the focus is on the timing e.g. right now, at this very moment
or to describe an action in process/ on the verge of happening but not complete (e.g 서고 있어요 ~ in the process of standing up 앉아고 있어요 ~ in the process of sitting etc.).
Note: in reverse, sometimes english simple present tense is used to represent korean present progressive too.
한국에(서) 살고 있어요 = I live in Korea.