The complaint about having a man's voice read a sentence with a feminine adjective referring to the subject is raised frequently in this discussion forum and many other discussion forums. Moderators and other users have pointed out every sentence is available in both a computer generated male voice and female voice. Think of it as someone quoting a sentence from a book instead of as a conversation with an individual. This approach actually forces you to pay attention to what is said instead of the speaker's gender in "Type what you hear" problems.
It's not so wrong, and perhaps Duolingo will add that to its list of acceptable answers.
In this context, saying I'm sorry and I apologize are equivalent; still, in other situations they are not. For example, if you are ill it's more appropriate to say I'm sorry I'm ill, than to apologize for being ill.
Another thing to consider is the learning process. Duolingo presents a set of words, phrases, grammar and context, and then gives us practice in using those tools. There will always be more nuances to learn later, either within Duolingo or elsewhere.
"Ocupada" (busy/occupied) is a Spanish past participle being used as a predicate adjective in this sentence. That's why, depending on the sex of the speaker, the gender of the participle changes. In English, subject complements are nouns that define the subject further, or they are adjectives that describe the subject. For example, "I am Linda/Soy Linda" has the subject complement "Linda," and "Estoy ocupada/I am occupied" has the adjective complement "occupied." "Soy" is used because being "Linda" is my identity, and my identity is permanent. "Estoy ocupada" is a progressive activity, and "estoy" is used because, presumably, I will not be busy forever and thus my "busy-ness" is a temporary condition.
In theory, you would say "Soy ocupada" only if you were speaking of something you do that you will never stop. For instance, you might say "Soy preocupada con cambio climático/I'm worried about climate change." The inference would be that you will always worry about the climate. However, I think this is not colloquial Spanish and is something that native Spanish speakers do not do. They take the long view that permanence is very rare. That is why location and time use the Spanish Continuous Tense; even mountains crumble, and time is always passing.
As far as I know, Spanish past participles are used as adjectives, and Spanish present participles (gerunds) are used as noun substitutes, as in "Reading is my favorite hobby." (El leyendo es mi pasatiempo favorito.) In this sentence, the gerund is the subject, and most Spanish subjects require the article, so I'm thinking that perhaps a gerund subject always needs the article as a cue that the first word of the subject is acting as a noun substitute and not acting as a verb. I have not seen Spanish "-ing" (-ando, -endo, -iendo) words used as objects. Instead, English gerunds seem to be more easily and more often translated into Spanish infinitives when the English sentence has a verbal being used as an object (Verbals are defined as English infinitives or English -ing words used as another part of speech). For example, "My favorite hobby is to read." (Mi pasatiempo es leer.)
"Lo" doesn't have to do with the word "I" in English. "Lo siento" is a phrase meaning "I'm sorry", but it is literally translated as "I feel it". "Siento" means "I feel", and "lo" can be put before a verb to describe the verb as happening to "it".
TLDR: It's confusing, but it would be easier to remember "Lo siento" as the phrase "I'm sorry". Estoy, by the way, means "I am".
Lo siento means I'm sorry. Perdón means pardon or excuse me. I'd use it whenever I'd say pardon or pardon me in English. Disculpe is excuse me and I'd use it when I'd say excuse me in English. They are somewhat interchangeable I think. I don't know what the last one means.