1)  Women may have progressed in the Church, but they are not treated as equal members of the Body of Christ.  
2)  Perhaps the Church ought to be more of a democracy than it is.  From what I have read, the early Church was far more democratic than the institution that we have today.
3) In what sense would it not be "proper" to have a woman Pope?  Just to allay your fears, even if the Catholic Church were to permit the ordination of women (as I believe it should), it will take many generations to have a woman Pope, so your sensibilities will, most likely not be offended, because you will probably not live to see the day.
4) Yes, I happen to believe that it is not right that gays are not afforded the Sacrament of Matrimony.