It is extremely common. Korean is a very contextual language, so if it is obvious what is being referred to, it is often (maybe even "usually") not spoken. This is not only true about subjects, but also often true of the particles.
This is somewhat problematic for a course like Duolingo, since single sentences without context can have a wide variety of meanings. Depending on the situation, "아뇨, 괜찮습니다" could mean:
No, it's fine /OK / all right
No, he / she is fine / OK / all right.
No, we / they are fine /OK / all right.
No, I am fine / OK / all right.
The subject could actually be almost anything you can think of that might have been under discussion. "아뇨, 괜찮습니다" could be an appropriate response to all of the following questions / statements.
Shouldn't the car be moved?
Is your daughter hungry?
Is the food too spicy?
Shouldn't the dog go to the vet?
Sorry to be so rude.
Apologies for taking too long.
Are you busy?
people often say something that sounds like "갠찮아" but the way it's spelled does not sound EXACTLY the same, although very similar. also, don't spell it "...슴니다" because that's not proper (although in korean if you change one thing or even do anything to a word you can just call it slang ex. 진챠, 안뇽, 조아, 응, 고마오, 기요미, etc.)
both should be correct. a lot in the korean language depends on the context of the conversation. if A asks wether B is hurt and B answers 괜찮습니다 the correct translation is “I'm okay“. on the other hand, if A asked wether B needs help with for example carrying shopping bags or how B likes a certain book and B answers 괜찮습니다 “it's okay“ is the correct translation.
Well, because Korean is a high-context language, you usually leave out the pronouns, because it is clear from the situation about whom you are speaking. Because No, I am okay is clearly is an answer to a question which made it clear about whose okayness was being asked, the I is left out in the answer.