Robusto's response is fantastic and very informative, but I noticed that it failed to answer your question directly in relation to the two scenarios you shared. So, let's see...
In the case of the first conversation, "who cares?" is used in a way that is to my mind reflective of a dismissive, arrogant, and disrespectful disposition. And it's unprofessional to say the least. The fact that you came here in the first place to ask about it already implies that this phrase was used in a manner that made you uncomfortable, which would be consistent with my assessment. I would say that your discomfiture is warranted. I myself would be wary of someone who expresses himself this way about the needs and concerns of clients in a professional environment.
He could have said something like, "We don't have the resources to address the needs of 300 clients. They're going to have to endure for now. We'll revisit this issue if and when those resources become available." But this requires a little more patience, a bit more intellectual cohesiveness, and a degree of respect for both the clients and those present. The dialog you've shared lacks all of these. So there's that.
In the case of the second dialog, "who cares?" implies a complete lack of concern for certain social mores. It also implies a lack of concern for what others think, feel, or believe about this particular behavior. He wants to be free to pick his nose without being bothered about it, so he very clearly states his position so as to cut-off any further debate, "Who cares?" That is to say, "You're the only one that seems to care, so I'm not about to change what I see as a perfectly innocuous behavior just for you." So, the use of the phrase in this particular context means he wants you to back off and drop the subject. It's not open for discussion. He'd rather you look the other way if he's got his fingers up his nose rather than watch you make faces, look horrified, and tell him where and when he should have his fingers up his nose. It's possible that he has dealt life-long with tormenting sinus issues that have ultimately driven him leagues beyond caring what people think about where, when, and how he deals with his sinuses.
This does not necessarily imply arrogance or disrespect--he may be quite the team player and he may otherwise show respect to his peers. He may deal with a concern such as that posed in the first dialog that demonstrates a degree of respect. On the other hand, the speaker in the first dialog may be horrified to be caught with his fingers up his nose and immediately rush off to the bathroom without another word.
So both uses of the phrase imply a degree of disregard, but in very different ways. The first is a clear breach of professionalism while the latter is more the result of a clash of cultural and/or personal values.