So I Google the definitions:
Emotion :any strong feeling; associated with mood, temperament, personality, and disposition.
Feeling seems to have a wider scope:
Feeling: experiencing of affective and emotional states; a vague idea in which some confidence is placed; a physical sensation that you experience; an intuitive understanding of something; emotional or moral sensitivity.
hmm...
Basically, do you let a feeling create an emotion? Ie. Feeling hopeless leads you to being emotionally sad, distraught, and angry...
Can you separate Emotion from feelings? It seems that the core definition of an emotion is the that it is the consequence of a feeling.
It might be very easy to hide an emotion if the feeling is there. However, if the feeling isn't there initially, there should be no emotional consequence, and vice versa. The duration and significance of impact varies upon the person.
Is it easy to detach and not feel anything from a prospective issue/person, etc ? Yes.