Hi, thanks for using healthcare magic
The black dots in the stool may be dried or old blood. Blood can be seen with inflammation or infection of the upper respiratory tract
, it should resolve with the infection.
associated with a viral upper respiratory tract infection can sometimes last up to 6 weeks in some persons.
In your case the persistent nasal discharge and post nasal symptoms would be contributing to your cough. You may want to consider the use of a topical steroid nasal spray
and an oral antihistamine. A decongestant would also help, these often may be found in combination with anti histamines
If the symptoms persist you should consider visiting an ENT doctor for evaluation
I hope this helps