Could you please back this up with the hadeeth to make things clearer and also please try to give full reference of the hadeeth to everything you state ?

(But with regard to the prayer (women praying the funeral prayer), there is nothing wrong with that. Women may join the funeral prayer. The prohibition applies only to visiting the graves. Women should not visit graves according to the more correct of the two scholarly opinions, because of the ahaadeeth which indicate that that is forbidden. She does not have to offer any kafaarah, all she has to do is repent.)   

Can you tell me if women prayed Funeral Prayers in the times of our prophet(P.B.U.H)?

Praise be to Allaah. 

Praying the janaazah (funeral)
prayer is prescribed for both men and women, because the Prophet
(peace and blessings of Allaah be upon him) said: “Whoever attends the
janaazah until he offers the prayer will have one qeeraat
(of reward), and whoever attends until (the deceased) is buried will
have two qeeraats.” It was said, “O Messenger of Allaah, what
are the two qeeraats?” He said, “Like two great mountains,” meaning,
of reward. (Saheeh – agreed upon). But women should not follow the funeral
procession to the graveyard, because they are not allowed to do that,
as it was reported in al-Saheehayn that Umm Salamah (may Allaah
be pleased with her) said: “We were forbidden to follow the funeral
procession but it was not made absolute on us.” (narrated by Muslim).
But women are not forbidden to offer the janaazah prayer, whether
it is offered in the mosque, in a house or in a prayer-place. Women
used to offer the janaazah prayer with the Prophet
(peace and blessings of Allaah be upon him) in his mosque and after
his lifetime. Visiting graves, however, is something which is only for
men, as is following the funeral procession, because the Messenger
(peace and blessings of Allaah be upon him) cursed women who visit graves.
The reason for that – and Allaah knows best – is that there is the fear
that if women were to follow funeral processions to the graveyard or
visit graves, that would cause fitnah (temptation) to others
or to themselves. The Prophet (peace
and blessings of Allaah be upon him) said, “I have not left behind any fitnah more harmful to men than women.” (Saheeh – agreed upon). And Allaah is the Source of strength.