In the book "Psychologie" by Richard J. Gerrig, Philip Zimbardo I read there are studies claiming that:
"die gegenwärtige Forschung legt nahe, dass das Hormon eine facettenreiche Rolle bei persönlichen und sozialen Prozessen spielt" (IsHak et al., 2011)
Translation: Recent studies suggest that the hormone plays a diverse role in personal and social processes
Another study was carried out in order to examine the effects of oxytocin on trust. In this study, half of subjects where given a dose of oxytocin, and the other half not. Furthermore, they carried out fMRT scans to examine the areas on which the oxytocin had an effect on. The study showed that:
die Scans ergaben, dass Oxytozin-Teilnehmer weniger Aktivität in GehirnRegionen aufwiesen, die wie Amygdala mit Furchtreaktionen in Zusammenhang stehen. (Baumgartner et al., 2008)
Translation: The scans suggested that participants who were given a dose of oxytocin showed less brain activity in regions which are, like the amygdala, associated with fear.
Unfortunately, I do not have access to the paper where this seems to be described (Baumgartner et al. 2008).
To what extent does oxytocin reduce brain activity and in which regions?
Baumgartner, T., Heinrichs, M., Vonlanthen, A., Fischbacher, U., & Fehr, E. (2008, May 22). Oxytocin Shapes the Neural Circuitry of Trust and Trust Adaptation in Humans.
The dreaded it depends.
This question was discussed behind the scenes in chat and we concluded with the consensus on explaining why this question is hard to answer.
Basically, you are asking on the suppressive effects of oxytocin on the brain.
First, it depends highly on the brain area under investigation. Many, and perhaps all imaging studies focus on a few promising structures in the brain. Those promising structures (regions of interest) need to be identified by prior research. The effect on amygdala and other fear centers has already been answered in the question (suppression). So to answer this question, one has to review all the existing literature to find any brain structure that is suppressed. This community is not here to conduct literature reviews, or write review papers. We're here to concisely answer focused questions, as outlined in the help center. As written there, the why and how questions are great. The what, where and how much questions like this are questionable questions.
Second, suppression is relative. If you investigate a subject in a certain state (e.g., terrified, when experiencing feelings of love or hatred) and test a drug such as the love hormone oxytocin, it may, or may not have totally different effects. The physiological state is very important. Suppressive actions can become evident in a hyperactive system, but have the opposite effect on a neural system that is relative inactive. It is all not so black-and-white. Hence - again - questions need focus and be outlined such that borders exist that emphasize, and most importantly limit the scope of the question.
In all, my answer is that it depends on the brain area under investigation and on the physiological state of that brain area.
Note. We do appreciate all your background research, your efforts in improving the question and I seriously do love neuroscience. It's my job, it's my passion, it's my hobby. Please feel more than free to further refine your interesting question and please also feel more than encouraged to continue to enrich this site with your contributions.