Clustering vs Non Clustering problems?
Both of the examples are clustering examples. Clustering is about grouping of similar dataset when one is not given the data.

In the gene problem,


One possible setting is you are given the DNA micro-array data. Your task is to learn how many types of people are there. This is an unsupervised learning problem, we are not given the labels. We just group  people with similar type of genes together. 


In the cocktail party problem:


There are two people in a room and there are microphones. We just record the audio and pass it to an algorithm and tell the algorithm, hey, learn the pattern if there is any. Each individual might have their own speech pattern/ language/accent. The algorithm pick it up on its own such pattern and detect that there are two people in the room and they can distinguish the speeches of the two people.


Let me give you an example of a non-clustering example. 

For example, a question of interest is to detect anomaly. For example, what you are given could be thr normal operating state of a machine, say their sensor readings. From those readings, you have to learn what is normal for the machine and you have to figure out when you are given a new data point, should it be considered normal.