Although assessing the colour of the mucus isn't all that reliable, it's something your doctors may have considered.
http://abcnews.go.com/Health/Wellnes...6427491&page=1
Green mucus usually means there's an infection, often bacterial or the flu and usually causes sinus pressure build-ups, however, if it's been lasting for a month, using the mucus colour grading, a likely candidate is bronchitis. It's easily confirmed by a throat swab.
You may want to get aerosol-inhaler bronchodialators for short-term relief when coughing gets pretty bad. They're meant for asthma but they're also used for infected bronchi-bronchioles.
The blood is probably a by-product of all the coughing. About 4 years ago I had bronchitis and it's not something that you can quickly recover from over a weekend filled with rest. It took 1-2 weeks (maybe 2.5 weeks) to recover. When it got particularly bad, I was "coughing up" blood, not just a little droplet but a decent amount that made me think I was soon going to have my lung on the bedroom floor. I went to the doctor and he assured me it was just due to a shitload of coughing.
Not to be too nosy but is there something in your medical record that suggests you have confabulated symptoms in the past? No half-decent doctor would tell their patient, "you're clean as a whistle" when they look dead tired, coughing away and have dribbled a bit of blood out from their throat. It's odd for one doctor to turn you down but if TWO turn you down, either you have bad luck and stumbled upon uncaring lousy doctors or there's something big you haven't mentioned from your medical record.