1) If in your society it’s seen as supporting Christian beliefs, then it would be haram to wear it. I would say that in most societies it would be viewed by people as supporting Christian beliefs, so you should not wear it.
2) If this symbol is a pagan symbol, then wearing it is basically supporting pagan beliefs in one way or another, so one should not wear them. Yes if the symbol is completely neutral and people in society don’t see it being associated with any pagan beliefs then it would be ok
3) The Shi’a traditions don’t use the Hamsa symbol. They simply use the hand symbol to refer to the cut hands of al-Abbas (a). It’s clearly known by society that this is what it refers to. Hence, it doesn’t support any pagan beliefs or traditions
4) Yes if the Fravashi refers to false Zoroastrian beliefs then one cannot keep it at home