Can a man who had only one wife and is now divorced remarry? Does the rule vary if they do or don't have children? If yes, how? If it's allowed, is it compulsory or optional?
Allah says in Surat Al-Baqarah (interpretation of meaning) :
And divorced women shall wait (as regards their marriage) for three menstrual periods, and it is not lawful for them to conceal what Allah has created in their wombs, if they believe in Allah and the Last Day. And their husbands have the better right to take them back in that period, if they wish for reconciliation. And they (women) have rights (over their husbands as regards living expenses) similar (to those of their husbands) over them (as regards obedience and respect) to what is reasonable, but men have a degree (of responsibility) over them. And Allah is All-Mighty, All-Wise.
A woman should wait for at least 3 months before marrying someone again. There is no compulsion for man or woman to marry or not marry after divorce. Although getting married is recommended by Prophet (P.B.U.H)
From Sahih Bukhari
Narrated 'Alqama: While I was walking with 'Abdullah he said, "We were in the company of the Prophet and he said, 'He who can afford to marry should marry, because it will help him refrain from looking at other women, and save his private parts from looking at other women, and save his private parts from committing illegal sexual relation; and he who cannot afford to marry is advised to fast, as fasting will diminish his sexual power."
The law of Islam has created several solutions and lines of defense for family before resorting to divorce ...
وَاللَّاتِي تَخَافُونَ نُشُوزَهُنَّ فَعِظُوهُنَّ وَاهْجُرُوهُنَّ فِي الْمَضَاجِعِ وَاضْرِبُوهُنَّ ۖ فَإِنْ أَطَعْنَكُمْ فَلَا تَبْغُوا عَلَيْهِنَّ سَبِيلًا ۗ إِنَّ اللَّـهَ كَانَ عَلِيًّا كَبِيرًا
As for women you feel are averse, talk to them suasively; then leave them alone in bed (without molesting them) and go to bed with them (when they are willing). If they open out to you, do not seek an excuse for blaming them. Surely God is sublime and great. [An-Nisa:34]
إِنْ خِفْتُمْ شِقَاقَ بَيْنِهِمَا فَابْعَثُوا حَكَمًا مِّنْ أَهْلِهِ وَحَكَمًا مِّنْ أَهْلِهَا إِن يُرِيدَا إِصْلَاحًا يُوَفِّقِ اللَّـهُ بَيْنَهُمَا ۗ إِنَّ اللَّـهَ كَانَ عَلِيمًا خَبِيرًا
If you fear a breach between them, appoint one arbiter from the people of the man and one from the people of the woman. If they wish to have a settlement then God will reconcile them, for God is all-knowing and cognisant. [An-Nisa:35]
And when impossibility of married life, the divorce is the last solution to keep this family, but for the divorce terms and conditions and controls, the Muslim should abide by it.
الطَّلَاقُ مَرَّتَانِ ۖ فَإِمْسَاكٌ بِمَعْرُوفٍ أَوْ تَسْرِيحٌ بِإِحْسَانٍ ۗ وَلَا يَحِلُّ لَكُمْ أَن تَأْخُذُوا مِمَّا آتَيْتُمُوهُنَّ شَيْئًا إِلَّا أَن يَخَافَا أَلَّا يُقِيمَا حُدُودَ اللَّـهِ ۖ فَإِنْ خِفْتُمْ أَلَّا يُقِيمَا حُدُودَ اللَّـهِ فَلَا جُنَاحَ عَلَيْهِمَا فِيمَا افْتَدَتْ بِهِ ۗ تِلْكَ حُدُودُ اللَّـهِ فَلَا تَعْتَدُوهَا ۚ وَمَن يَتَعَدَّ حُدُودَ اللَّـهِ فَأُولَـٰئِكَ هُمُ الظَّالِمُونَ
Divorce is (revokable) two times (after pronouncement), after which (there are two ways open for husbands), either (to) keep (the wives) honourably, or part with them in a decent way. You are not allowed to take away the least of what you have given your wives, unless both of you fear that you would not be able to keep within the limits set by God. If you fear you cannot maintain the bounds fixed by God, there will be no blame on either if the woman redeems herself. Do not exceed the limits of God, for those who exceed the bounds set by God are transgressors. [Al-Baqara:229]
First divorce (divorce small Baynunah): Is that the man called his wife by saying a one-time word: You are divorced or I divorce you, or other words that have the same significance.
The right of the man who divorced his wife a divorce samll Baynunah , can restore it when he wanted without Mehr, contract, or witnesses, that were not completed iddah (three whole months).
Not eligible for a man to recover his wife, who divorced her divorce Baynunah samll, if they exceed the period of three months, then so is shifting from divorce to divorce samll Baynunah to Baynunah big
Second divorce (divorce big Baynunah):
Is that the reasonable man called his wife the right three divorces in varying times, or be divorce her one divorce and ended iddah of the wife three months.
The children not connected to them to rule of divorce or reversed from a shar'i point, but in some cases be continuation in married life itself is a harm to children and family, and the divorce is the final solution for the benefit of the Muslim family when impossible to married life.
But for a Muslim to adhere to conditions of the divorce and his sunnah as not to falls in forbidden divorce