Can I omit certain words if it is understood what I am talking about?
First, in English word omission is quite common, especially in short conversations and answers:


  &ndash; Is he an engineer?
  &ndash; Yes / Yes, he is / Yes, he is an engineer (too redundant).


In Chinese, however, word omission is even more common, as you've suggested. With my living experiences in mainland China (native Chinese), I can tell that both the full versions and the shortened versions are right, while shorter versions are more common in oral Chinese.

Example:


  &ndash; 你最近过得好不好？
  &ndash; ((我最近)过得)好。  




Another notice: In Chinese, when asking about feelings, people don't respond with "I feel / I don't feel", but the actual feeling instead. So your last example should have been


  "你感觉开心吗" "(感觉)不开心"


People very rarely reply with 感觉 without the actual feelings.