Should we treat it as a holy day of obligation? Can we work on Good Friday? Should this day be a day of sadness for Catholics?
A holy day of obligation means you are required to attend Mass on that day. Since there are no Masses on Good Friday you couldn’t treat it as a holy day of obligation even if you wanted to.
As for can you work, I guess it depends on what you mean by work. Do you mean go to you place of employment? I think that depends on the job. Are you a policeman, doctor, paramedic, or in some other profession that serves a critical need in the community? If yes and you are scheduled, go to work. If you mean can I do housework or some such thing, then you need to know that there is no prohibition against doing such things at any time. My personal preference for Good Friday (and Sundays, for that matter) is do what needs to be done today ( a load of laundry for Monday, cooking dinner, etc) and leave other things for other days.
Should we be sad? Sadness is an emotion, so there is no “should”. Good Friday is, however, a very somber time. We should refrain from doing frivolous things, seeking entertainment, socializing, etc. It is a day to be spent in prayer, silence, contemplation. That said, I will turning off the computer and TV shortly. I spend my Good Friday during the Passion hours reading the Gospels.