I'm back. I'm sorry that your question wasn't answered in a timely manner. It's difficult for us to jump in the middle of such a complicated case that has already been attended to by specialists - in this case at the Dick Vet Small Animal Practice. I'll address your concerns as you posted them...
The very dark stool is often associated with digested bood in the stool but slowed motility of the GI tract is the more likely reason for the darkness. The bacteria in the GI tract metabolize bile into stercobilinogen which gives stool its characteristic color. When motility slows those bacteria have more time to work on the bile and so the stool becomes darker.
We can still aspirate salivary glands although I admit that such a diagnostic isn't likely to be helpful in Papi's case.
Increased vagal tone due to GI inflammation/hyperactivity can slow the heart. It isn't clear to me if that's being termed an arrhythmia or another type of arrhythmia exists in addition to that caused by increased vagal tone.
A nonregenerative anemia (Papi isn't able to efficiently replace the red blood cells he lacks) is consistent with any chronic inflammatory disorder. It doesn't clarify which disorder is present, however. The enlarged lymph nodes in his abdomen may be reactive rather than malignant as would be the case with the malignant lymphoma. In other words, draining areas of inflammation and reacting to that activity.
Vomiting may be the only symptom of inflammatory bowel disease - particularly in cats. Please note that IBD can nauseate and result in vomiting. In addition, his swallowing excessively also indicates nausea and gastroesophageal reflux (GERD, acid reflux, heartburn). This should be addressed with an over the counter antacid such as PepcidTwo dosed at at 0.25 mg/lb twice daily of the famotidine component of that antacid.
Finally, only scoping and biopsy of Papi's GI tract might reveal the etiology of his vomiting.
Please respond with further questions or concerns if you wish.