Hi there, I'm Dr. Sara. I'm a licensed veterinarian who works exclusively with dogs and cats. I'm sorry to hear that Otto is having some health problems. I know that it can be frustrating when they are ill.
Hair loss is a fairly nonspecific indicator of disease. One common cause is allergies - but usually they have allergies starting early on in life that progress and become more severe as they age. Allergic disease can cause severe itching and secondary yeast or bacterial overgrowths that can compound the itching and cause hair loss. Another common cause for hair loss in older pets are hormonal issues like hypothyroidism or cushing's syndrome. The hormonal diseases usually look a little different on physical exam than the allergic ones (although not always), so if I have a suspicion for hormonal diseases I check a blood panel that would include the thyroid level, red and white blood cell count, and chemistry panel.
Actually all of the symptoms that you've described can also be attributed to the prednisone. Prednisone can easily cause hair loss, heavy breathing, panting, and difficulty jumping up (from overall muscle weakness). So if these signs started or became worse after you began the pred, it's like the prednisone causing the issues. Hypothyroidism can cause hair loss and decreased exercise ability, but not usually heavy breathing or panting.
A reluctance or inability to jump up onto furniture could either be due to pain (such as spinal/back pain, which is a common issue in dachshunds!) or muscle weakness. While spinal pain is treatable usually with pain medications and anti-inflammatories, the muscle weakness is a very vague symptom that would require a little more digging to diagnose properly.
Green boogers in the eyes can be a sign of simple conjunctivitis (pink eye), or possibly dry eye. I also see plenty of allergic dogs get conjunctivitis as well, so it could be another symptom of the main problem. Steroids like prednisone make them a little more prone to infections, so it could be that he just has an eye infection. A couple of simple eye tests can sort out the cause of his eye problems.
Cushing's Syndrome is a combination of clinical signs caused by excess steroids in the body - most commonly excessive thirst and urinations, excessive hunger, hair loss or recurrent bacterial skin infections, thinning of the skin, and "pot belly" appearance. The prednisone he's taking is a steroid, so it could be causing all of these things. If these signs started before the steroid, he could potentially have Cushing's disease, which is where the adrenal glad is producing too much cortisol (the body's steroid hormone).
I would suggest having him to the vet to investigate his eyes (exam, tear production test, pressure check to rule out glaucoma, and a fluorescein stain to check for corneal ulcers) as well as to discuss running some blood work on him to sort out the cause of the signs that you're seeing.
Please let me know if I can answer any follow up questions for you :)