I'm sorry that your question wasn't answered in a timely manner. We don't have many avian vets on this site. Both his cough/sneeze/hiccup and feather pulling will need the attention of an avian vet (please see here: www.aav.org). I suspect that the cough/sneeze/hiccup represents respiratory infection perhaps with an underlying vitamin A deficiency. This is treated with an antibiotic and vitamin A supplementation. It's important to note that once an African Grey acts ill they're quite ill. This is a protective mechanism because sick birds are attacked by other birds in the wild. Apallo's symptoms are important but they're not pathognomonic (specifically indicative) of any one disorder.
An avian-oriented vet might first treat symptomatically and supportively by providing supplemental fluids and electrolytes by needle and tube feeding a "recovery" food. Blood tests and cultures of Apallo's choana - the slit between his oral cavity and nose - and cloaca (vent) may be taken.
Until Apallo can be attended to, please heat up his environment to 85F by means of a 100W bulb shined into his partially covered cage (not at night when he needs to rest) or by taping a heating pad set on its lowest setting to the sides of his cage. If he appears weak, remove his perches and put his food and water on the bottom of the cage along with him. Add a water soluble avian vitamin such as Oasis brand to his water at half of the recommended dose so as not to make his water distasteful. Add a calcium supplement such as Calcivet or Calciboost to his water. These supplements are available in pet/feed stores. Avoid over the counter antibiotics designed to be placed in his water. They won't be effective if only because an ill bird won't drink enough to medicate itself properly.
Nutritional imbalances are a common cause of illness in our pet birds. What has Apallo's diet consisted of, please? Seeds should compose less than 20% of his diet. Ideally, a balanced pelleted diet such as can be found here: www.harrisonsbirdfoods.com or here: www.lafeber.com/pet-birds should be fed as well as hard boiled egg yolk, pancakes and cornbread, the tops of fresh greens, dairy products such as yogurt and cheese, fresh fruits such as apples, pears, melon, kiwi, and berries, vegetables such as broccoli, carrots, beets, asparagus, cabbage, sweet potato, and squash, and even tiny pieces of meat.
His feather plucking is particularly vexing. The etiology of feather picking/chewing/self-mutilation can be a challenge to determine particularly if you don't have an avian-oriented vet available where you live. Both medical and behavioral causes exist.
Birds usually chew the breast feathers and areas under the wing and around the legs but any pattern can present. Feathers on the head remain unless a molt or medical disorder is present. Medical disorders include viral infections such as psittacine beak and feather disease or polyomavirus, bacterial folliculitis, dermal parasites, yeast infection, and topical irritants need to be considered as well. Rarely do birds feather pick due to external parasites. Less commonly, infection with the protozoan parasite Giardia spp. will cause picking of the feathers over the ventral (lower) abdomen and heavy metal ingestion such as zinc can cause a feather-picking toxicosis due to chewing on zinc-plated cages. Anecdotal reports suggest that a unilateral (one-sided) pattern of picking may indicate a disease process under the area of picking such as ovarian or renal disorders.
There are many rule-outs for behavioral feather picking including improper socialization when raised by humans resulting in phobic birds, a "nervous" bird as you described or those with obsessive-compulsive disorders - which should be a consideration when this behavior is preceded by a molt. A traumatic event can cause a bird to become nervous and pick; anecdotal examples abound such as witnessing an attack by a hawk outside the window at a bird feeder, the owner leaving for vacation, a change in the color of the cage, a nervous owner, and the death of a mate or owner. Some birds improve in a new home with a new owner for unknown reasons.
Here's where an avian-oriented vet will be necessary...
All possible medical causes for a bird's feather picking are evaluated first; then if no medical cause is found, behavioral causes are explored or presumptively treated with psychotherapeutic drugs such as clomipramine which would need to be prescribed by a vet. A complete blood count and biochemical profile, blood lead and serum zinc test (Louisiana Veterinary Medical Diagnostic Laboratory can run this test on just 0.1ml of blood and serum), X-rays, psittacine beak and feather disease test, and fecal ova and parasite exam and fecal ELISA test for Giardia spp., if necessary, are performed along with a feather follicle biopsy and culture. Consider the introduction of toxins into Apallo's environment as well - the great majority of which can’t be tested for.
General and presumptive treatment includes removing any stressors, improving his diet and restoring what he perceives as a normal environment. Consider clomipramine which works well in true obsessive-compulsive disorders. Distracting Apallo with toys and a "sweater" over the area may help but be sure that he can also engage in normal and necessary preening behavior. A collar isn't recommended as it doesn't allow him to engage in normal preening behavior, normal feeding behavior, and normal movement. A collar should only be used if he's in imminent danger of hurting himself (self-mutilation). Finally, many birds normalize once their molt is complete and so simply "watchful waiting" may be appropriate at that time.
Please respond with further questions or concerns if you wish.