Yes, no, and not quite

English makes 'is not' by adding not to is. Korean has a separate verb, 아니에요, and the noun it denies takes the 이/가 marker you met on day 4: 'American person, is not'. That is why the full sentence is 미국 사람이 아니에요, though in quick speech the 이 often drops. Spell it 아니에요, never 아니예요: 예요 only follows a noun ending in a vowel, and 아니 is not a noun but the stem of the verb.

괜찮아요 does three jobs because of what it says, roughly 'it is no problem'. In reply to an apology it means no harm done; about yourself it means I am fine; in reply to an offer it means there is no need, which is how Koreans decline politely without the bluntness of 아니요. The situation tells you which, and a small smile with a raised hand makes it a refusal. 맞아요 is different: it comes from a verb meaning to fit or be correct, so it confirms facts and never accepts offers.

몰라요 is not 'not know'. It comes from a separate verb, 모르다 (moreuda), whose whole meaning is to not know, rather as English has ignorant beside aware. Said flatly, 몰라요 can sound curt, like 'no idea'. 잘 모르겠어요 adds 잘, 'well', and the same -겠- that softens 처음 뵙겠습니다 on day 2, turning it into something like 'I could not really say'. That small cushion is why it is the polite choice with anyone you do not know.