First, I would just note that the watery eyes and nose (where the black is just likely hyperpigmentation related to nasal irritation) is likely just a side issue. We can see opportunistic viral agents cause minor signs when the immune system is being challenged by something else. Therefore, as long as those stay clear (where snotty/yellow/green is a sign of bacterial infection and an indication for antibiotics), we'd just want to monitor those +/- flush his eyes with sterile saline (ie OTC first aid eye wash, plain contact lens solution).
Now in regards ***** ***** diarrhea, as he cannot tell us what we ate, we can only assume this is our culprit. Still it was reasonable for the vet to worm him just in case we had a parasitic cause. As well a light diet as you mentioned is good. Other options that you could also use include pasta or boiled potato in place of the rice. And you can also substitute the turkey for boiled chicken, white fish, scrambled egg, or cottage cheese if you wish. As well, you can also add a spoonful of tinned pumpkin or all bran to whichever you feed to bulk up his stools.
Though further to this, I would note that since the blood is scant (and likely just secondary to the diarrhea irritating his colon), you could also use a dog safe anti-diarrheal. In this case, you could use Kaolin/Kaopectate (More Info/Dose @ http://www.petplace.com/article/drug-library/library/over-the-counter/kaolin-and-pectin-kapectolin-k-p) from your local pharmacy. Or you could alternatively use a Kaolin/probiotic for dogs like Fast Balance, Propectalin, or Protexin Prokolin. All are available over the counter at your vets, pet stores, and even Amazon. And these just help slow the abnormal stool and just make sure he doesn't loose any more fluid or nutrition then he has to.
Finally, if you use those alongside with your vet's care, we will hopefully see him settle. If we don't for any reason, do consider submitting a stool sample. This can be sent for testing to tell you what is actually causing the diarrhea (ie protozoa, parasites, bacteria, viruses) and that can ensure you identify anything infectious if present and target treatment to properly clear them for your lad.
All the best,