Question: I’d like to know how a lady who has been raped should be dealt with. Does this incident make her impure in any way? Does it make her an unbeliever? Is she guilty of adultery? If not, then is it permissible under Islam to marry her and give her a good life; but then the Qur’an says that a Muslim should marry a chaste woman only? 
    Answer: Rape is one of the most heinous crimes a man can commit. Obviously, his victim is absolutely blameless. On the contrary, she deserves all the sympathy and compassion a society can give her. A raped lady does not become impure in anyway. Only a willful indulgence in such an act makes the concerned man and woman impure. So, yes a chaste Muslim can marry her. 
The one thing that a person must consider is pregnancy; if the lady has become pregnant as a result and has passed the abortion period^*, then of course till childbirth she should not marry anyone. After childbirth only marriage options can be considered. 
You can now marry her keeping the implications of such a marriage in mind; you maybe doing a pious gesture, but your family and the society might make your life miserable; embark on this venture only if you are able to handle these pressures. 

^*. After conception, once the spirit has been blown in a human embryo and it becomes a living soul, it cannot be aborted unless on some medical grounds a doctor advises so. Before this spirit is blown, the human foetus is actually a ‘non-living’ mass of flesh and can be destroyed. A Hadith tells us that it is after 120 days (four months) that an angel comes and blows the spirit in the human embryo:

Indeed the creation of each one of you is brought together in the mother's belly for 40 days in the form of a sperm, then he is a germ cell for a similar period, then an embryonic lump for a similar period; then there is sent an angel who blows the breath of life into him... (Bukhari, Kitabu’l-Qadr)

    If this Hadith is authentic, and there seems no reason to believe otherwise, then abortion after 120 days can only and only take place if a doctor advises it on medical grounds.