Yes, it is permissible for her to wear it, as long as the script of the Quran does not touch her skin if she does not have wudhu'. To avoid this, she can wear it on top of clothing, or keep the script to the front, or cover it with a thin transparent layer of nail-polish, or something like that, which can also be used as a protection for it as well.
Of course, she must also ensure that it stays away from any najasah.
No friendship can be permissible between non Mahram men and women. Dealing with doctors or teachers or lawyers or government officers etc from other gender is permissible as far as it does not have emotional impact at all, which in fact not a friendship but just work only.
Friendship between non Mahram men and women is not permissible even if you have no emotional intention now. You can not guarantee the intention of the other person neither now nor in the future.
We believe in Allah's absolute Wisdom and Knowledge and Mercy,b that is why we never argue with any rule from Allah. We know that He Knows all the reasons even if we don't know. Real believers submit to all the rules which came from Allah (SWT) through His Messenger (SAWA).
We might know part if the reason but can never expect knowing full reason because of our limitations as human beings.
Woman covering her body might be related to the fact that woman in her nature is more attractive to men, that is why she needs to cover herself.
Woman is allowed one husband only at the time, because of her dignity and respect which part of her creation as a human female. Usually, noble females from any faith or religious or cultural background, don't accept nor digest having more than one husband at the time. That is part of her nature.
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