Should I stay home from work if I have copd?

Not necessarily... Many patients with COPD are able to work for many years after their diagnosis but it does depend on some factors like how physical is your job, how is your oxygen level and many other factors. Your doctor can determine if you are still able to work or if your disease is severe enough that you can no longer work at your current job. But, COPD does not necessarily mean you can no longer work.
No. A qualified "no". COPD causes shortness of breath which can limit one's ability to work, more or less depending on the job. That being said, most people with COPD can function in the workplace, with the exception of those exposed to fumes, dusts, particulates and other respiratory irritants.
No. It depends on what stage of COPD you have. Like with all chronic illnesses, you can have good days and bad days (exacerbations) with copd. It is important to rest and get medical treatment when you have these exacerbations, but overall you should try to stay active.
No. You need to consult your physician if you are having what is called a COPD exacerbation, which is a worsening of symptoms. However, just because you have COPD doesn't mean you have to stay home from work.