This is my view of the connotative meaning - Sadness: the most general "sad" emotion of these bunch Grief: sadness usually felt after the loss of something, such as a death. Sorrow: I see sorrow as in between sadness and grief. It's not as powerful as grief but holds more emotion than sadness. You may feel sorrow after forgetting something, hurting someone's feelings, or failing. Distress: This is tied to a situation that is ongoing and is likely in combination with anxiety or possibly pain. If you are running late, in possible danger, or under extreme stress you may feel distress. Anguish: Somewhat similar to distress, anguish is also related to pain except this time more intensely. It's usually associated with severe mental or physical pain/suffering. If you've just been seriously injured that following emotion could be described as anguish.

It can mean both. If I heard that sentence alone I would likely think someone meant that they will be staying at home that night, but it could also mean the other in the example that a man or woman on a trip would tell their spouse as they were returning home.

Here it means the same as "for". In the service of means to "work for". A worker is in the service of their company. In your sentence, the management of emotions is working for the goal of well being. Mangement of emotions is the worker, and well being is the company.