Hmm. Where did you get the information used to generate your questions? If it was from a responsible source, then I probably can find a research paper on it.
BTW:If you use NIH or pubmed as the site to search, you get pretty good references.
I did that. I do not see very much to substantiate either assertion.
However. This not my area of knowledge. So I looked around further.
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3693622/
This is a review article - that means they did a lot of reading and tried to put together reasonable answers about testosterone and behavior. It does not seem to have much to assert there are answers to your set of questions. There is variation in levels of testosterone secretion in men. Women normally have testosterone at much lower levels than do men, 15 to 70 ng/dL, which is a wide range. There are medical conditions in women, like Polycystic Ovarian Syndrome, in which women have above normal male hormone levels and may develop male traits like hirsutism. Which means they may grow facial hair, for example.
So what I am saying is there are varying physiological responses in men and women and ranges for normal hormone levels. And normal levels are broad and are age-related. Which means that 'teststerone-mediated' behavior is not simple.
The gonadotropin-releasing hormone (GnRH) is the primary "on-switch" to initiate the production of testosterone. Synthesis/release of the hormone does not appear to be related as a reaction to visual stimulus, AFAIK.
As far as the amygdala question, I pass. There are PF members who know much more than I do. Maybe
@DiracPool or
@Pythagorean might have something to say.