Answer
Salam Dear Mustapha,
Thank you for your questions and for contacting Ask About Islam.
Prayer is a relationship between Allah and the Muslim. In many verses of the Quran, Allah mentions Prayer and follows it with a mention of the remembrance of Allah. The Quran says what means:
{Recite what is sent of the book by inspiration to thee, and establish regular prayer: for prayer restrains from shameful and unjust deeds; and remembrance of God is the greatest [thing in life]. And God knows the [deeds] that ye do.} (Al-`Ankabut 29:45)
The importance of Prayer is so great that one is ordered to observe it even while traveling or when one is in fear, for it is a main pillar of Islam. The Quran says what means:
{When ye pass (congregational) prayers, celebrate God’s praises, standing, sitting down, or laying down on your sides; but when ye are free from danger, set up regular prayers; for such prayers are enjoined on believers at stated times.} (An-Nisaa’ 4:103)
So, we need to know that Prayer is not something that a Muslim does out of his or her own whim, but it was revealed from Allah to Prophet Muhammad (peace be upon him). Actually Prophet Muhammad said: “Pray as you saw me pray” (Al-Bukhari).
Also, Allah has ordered us to obey Prophet Muhammad and to follow him in everything because he is the role model of a perfect Muslim (a person who submits peacefully to Allah). Therefore, we cannot make additions to the acts of worship of the Prophet, for the Quran says what means:
{This day have I perfected your religion for you, completed my favor upon you, and have chosen for you Islam as your religion.} (Al-Ma’idah 5:3)
Also, Prophet Muhammad (peace be upon him) said:
“If somebody innovates something which is not in harmony with the principles of our religion, that thing is rejected” (Al-Bukhari).
Thus, this is the reason that we cannot invent new rules in the religion by our own desires. The Quran and Sunnah (traditions and habits of Prophet Muhammad) are the way to follow in order to achieve complete submission to Allah, and it was never mentioned in the Sunnah that the Prophet and his Companions used chairs in the mosques while praying. 
Prayer is a type of worship consisting of specific statements and actions. It begins by stating and pronouncing the greatness of Allah and is concluded with salutations of peace. Prophet Muhammad (peace be upon him) explained to us exactly how to pray, with all details.
Only the sick and elderly may use chairs in the mosques during prayers. All worshipers sit on the floor side by side in orderly rows and during certain parts of their Prayer, they stand shoulder to shoulder, again in orderly rows.
The rich, the poor, the fair, the dark, the laborer, the honored, all sit and stand side by side, facing towards the Ka`bah in Makkah. Whoever comes early takes a place in the front rows and whoever comes late takes a place in the succeeding rows.
During the Prayer all people are equal before God. They stand side by side, sit side by side, and prostrate humbly side by side to the One and Only God. No discrimination is to be made between worshipers, who are all but the servants of Allah.
Even in Judaism and Christianity of today, there is no one evidence in their Books (the Old and the New Testaments) that Jesus or any other prophet prayed on a chair like what you see in churches and temples.
There isn’t any evidence that other prophets ordered their followers to pray on chairs. On the contrary, it is mentioned in their books that prophets like Jacob and even Jesus used to prostrate to Allah (God) while praying.
As for your second question, we need to mention here that the role of women in Islam has been misunderstood in the West because of general ignorance of the Islamic system and way of life as a whole, and because of the distortions of the media.
The Muslim woman is accorded full spiritual and intellectual equality with the Muslim man. She is encouraged to practice her religion and develop her intellectual faculties throughout her life. In their relations with each other, both are to observe modesty of behavior and dress, and follow a strict code of morality that discourages unnecessary mixing of the sexes.
Prophet Muhammad (peace be upon him) granted permission to Muslim women to attend mosques and pray standing behind the rows of men. He even advised the Companions by saying: “Do not prevent the female servants of Allah from going to the mosque” (Muslim). Also, he specifically told husbands: “When your womenfolk ask you for permission to attend the mosque, do not prevent them” (Muslim).
Of course this permission to attend mosques was on the condition that women strictly observed the various restrictions imposed upon them by the Shari`ah regarding dress and behavior. So, both men and women are allowed to pray in the mosque in the same jama`ah (group prayer).
But how? The Prophet explained how: When men and women are together in the mosque, we should have first lines of men behind the imam, then children, and then women.
This is the way Muslims used to pray behind the Prophet (peace be upon him), and, as I mentioned, we cannot invent new rules in our religion by our own desires, but we stick to what came in the Quran and Sunnah. 
Remember that the ritual Prayer involves bowing and prostrating. The wisdom of separating men and women during Prayers is to protect the hearts of the Muslims, both men and women, from any distracting ideas or evil suggestions that the devil may whisper into their hearts.
This is especially between men and women, in order to keep their hearts pure and sincere and busy only with Prayer. For the same reason, the mixing between men and women in general has special rules in Islam.
I hope this answers your questions. Thank you and please keep in touch.
Salam.