Is Doxycycline working against my bronchitis?

Yes, Doxycycline does treat bronchitis - 7 day course will be enough unless you have pneumonia, but Doxy still covers - so you should still see a doctor to listen to your lungs and possibly do chest xray. Also over the counter mucolytics like Mucinex or with a cough suppressant (Mucinex-DM) will help with symptoms. Post bronchitis cough can last up to two months, so don't be worried if this occurs, the infectious part has resolved.