TL;DR
While the image you link does a poor job at citing its sources, the brain images it contains are unlikely to be doctored. Each "normal brain" images is different because they were not obtained during the same task. And yes, most brain-related problems will produce different brain activations from patient and non-patients during tasks linked to that specific issue.
I am not a neurologist, but I do study brain scans regularly, as my education research team uses fMRI to study learning processes from a neurological standpoint.
From what I understand of your question, you have 3 issues, which I will try to answer as simply as possible.
1. Are these scans real?
For the first question, I'd say that these scans are most likely 100% real. I honestly see no reason to believe they aren't. But without the citation for the study it was taken from, I cannot tell you what they're supposed to mean. Obviously, they are meant to be proof that the brain of a neurotypical person is different than that of an individual with a certain diagnosis, such as ADHD. What this difference is and what it means are very difficult to extract from this image, since I do not know how it was obtained.
However, a literature review from Konrad and Eickhoff (2010) does suggest that there might be differences in functional connectivity between a normal brain and an ADHD brain, especially in areas traditionally linked to awareness, consciousness and attention, such as the posterior cingulate cortex and the precuneus. In laymen's terms, ADHD patients cannot activate some parts of their brains as well as other people in certain circumstances, because their neurons aren't as connected as would be expected.
This review produces no brain scan itself, but cites many studies that used fMRI, one of them possibly being the source of the image you linked. So while I can't comment much on these specific images, I don't doubt that they come from real research.
2. Why are the normal brain scans different?
For your second question, the answer is pretty straightforward. Yes, all those "normal brain" images are different, and it's totally expected. First, they didn't use the same subjects, so slight differences are to be expected. The images you see are usually the average activity of ~10-20 human beings, whose brain are all slightly different (in shape and activity), and thus must be normalized into a standard model. When you publish a paper using brain scans as data, you usually state which model you use and the characteristics of the subjects, which allows for a better interpretation of the data.
The most important difference, though, comes from the task used in these studies. When you study cognitive processes, you can't just put people into a scanner and extract results. You need to have these people actually do something worth studying (it costs about 500$ to scan someone, here). For example, Rubia et al. (2010) used a stop-signal task to measure the difference between ADHD patients and neurotypical individuals. Specifically, their task asked the subjects to press left or right, according to the direction of the arrows shown on the screen. Sometimes, however, a left/right arrow would be shown at the same time as and up/down arrow. In this case, the correct answer was to avoid clicking, something ADHD patients usually have trouble with as it requires them to remain vigilant for an extended period of time.
In a similar fashion, Wolf et al. (2009) studied ADHD using working memory task. Specifically, the subjects had to view a series of letters, some of which were highlighted, and commit to memory the letters that came AFTER the highlighted ones in the alphabet (you see a "B", so you memorize "C"). They were then shown a single letter, and had to tell if it was part of the letters they were supposed to memorize (they see AFGP, so if they then see B, H or Q, they must answer yes).
Both of these studies showed some differences between subjects with and without ADHD. Why do the normal brains in your image look so different? Just like in these 2 studies, simply because the brains weren't scanned during the same activity.
3. Can you really find different brain patterns for every psychic problem?
Finally, for your last question, one has to consider the history of brain research. Initially, brain scans were technologically impossible. What they had access to, however, were patients in asylums and those that lived through serious brain damage. In some cases, they could perform an autopsy of the patient's brain after their death. A lot of mental disorders were thus associated to lesions and injuries in specific areas of the brain. Over 150 years later, we study the same mental disorders and impairments using fMRI and other imaging processes.
While not every patient suffered from brain trauma, many of these conditions can be explained by insufficient brain development just as well. And interestingly, insufficient brain development in certain areas (such as the posterior cingulate cortex and the precuneus, just like Konrad and Eickhoff exposed) will often lead to the same symptoms as lesions and injuries to these same areas (such as ADHD in the previous example).
So basically, yes; if some problem is linked to the brain, it is very likely that we will observe different brain activations between patients and non-patients during tasks known to be more difficult for people suffering from this problem. For instance, ADHD patients usually have trouble with tasks related to attention, so scan resulting from such tasks will usually be different for ADHD patients than for regular people.
