Oh my gosh, I could have posted this myself a few weeks ago. My 14 month old had exactly the same thing...started with red eyes, into a high fever, no appetite, cough and runny nose. I had him seen twice and it is just a virus. The cough is just finally going away now. Make sure you keep him hydrated...that can account for the stools. Pedialight or milk is fine..not water as they need to electrolights. I can second what Margy said..this is a very very bad year for virus'. Lots of nasty things going around. Our Pediatrician told me 6 to 8 weeks can be very normal for these virus' to run there courses.
That being said, listen to your instincts. If you think there is more to this, have him seen. We are their advocates and personally, I always believe better safe then sorry.
Oh, also...is he teething at all? During all this my son was cutting his first molars and was drooling like crazy. That was part of what the cough was. At night while sleeping, the mucus would pool in his mouth and throat causing him to cough badly. I use the baby vicks (make sure it's the baby vicks) on his back and chest as well as keep a cool mist humidifier going in his room. It seems to help a little. Good luck.
Some nasty bugs' going round this year, make sure he is hydrated with fluids other than milk, milk is a food, water and juice is good,with the coughing getting worse I think a trip to the doctor is in order if it is getting worse. I am assuming at this age he is eating solid food , has he eaten anything with beets in or anything that may color the stool, he would also be off his food if he is coughing and has a stuffed' up nose . good luck