Pets in institutions are the 'in-thing' today and considered to be therapeutically beneficial. Perhaps the owner (Garbutt) was ahead of the times in his thinking. More likely, however, is that taking the dog was a means to get Ned to the asylum without causing distress/anger on his part. It's also worth remembering that these places were run for monetary gain. If taking the dog meant a few extra coppers in the bank, I'm sure it would have been agreed to.
With regard to understanding Ned's state of madness, it's necessary to filter off all present day knowledge of mental illness. Diagnosis was a different matter then and now. I'm sure that illnesses such as those defined in the greater groups of: psychoses (including such illnesses as Schizophrenia and Bi-polar disorders) and neuroses (including such illnesses as depression and personality disorder) were around at the time. However, they had not been fully identified or named. Some of the most common diagnoses encountered by me in old case histories (prior to 1953) are: severely retarded, simple-minded, feeble-minded, idiot, mentally deficient, of unsound mind, non compos mentis, lacks understanding, and incapable of reason. Feeble-minded or incapable of reason were applied to persons for a multitude of different reasons, some having no resemblence whatsoever to what we term mental ill health today. One of the most common I've seen is pregnancy outside of marriage! You would be amazed at the number of women in their 70s and 80s who were still sitting, now institutionalized, in mental hospitals around the UK in the 1960s, having been admitted some 50 to 60 years earlier simply because they became pregnant - without the help of a "husband"! That was enough to have you "put away", as was the common expression, and for many that meant " the further away the better". Mental illness, as we know it today, didn't need to be present at all in order to be comitted to an asylum.
It was interesting to see the drawing of Dunston Lodge Asylum which headed the article you provided. Founded at the beginning of the 19th century it has many of the hallmarks of the asylum recommended in the 1845 act. This act was really an ammendment to an earlier act (around 1810 if I remember right) which advocated county care. There were however too many loopholes in that act which allowed counties to circumnavigate the law and is one of the reasons why work didn't get underway with asylum building until mid 19th century when the 1845 act enforced the earlier act.
In the drawing, you can clearly see the move away from treating patients as prisoners.- which was the aim of the law. There are gardens for recreation and therapeutic work and what appears to be a farm adjacent to but outside of the asylum walls where patients could work and contribute to the running of the asylum by producing food., so was the occupational therapy of the era. I note also an enclosed courtyard where dangerous patients could be safely exercised, allowing even those a chance to see daylight and feel the weather on their faces. Many of these features are common to the later builds and marked an advance in the treatment and conditions of the asylums.