Praise be to Allah.
With regard to the first question, the answer depends on the circumstances. If what is intended by this action is to reduce the level of tension in the relationship, then they will come back together, or as a trial separation to see what effect that will have on them and their children so that it will help them to take a decision, and they both agree to this temporary separation, then there is nothing wrong with that.
If this is a decision that they have both taken, and have agreed to separate from one another without divorcing, then it should be said that if the woman foregoes the rights that she will lose through this separation, and he also foregoes his rights over her, and they think that this is in their best interests and those of their children, and the place where the woman and her children will stay is a safe place where they will not suffer neglect, then that is permissible, subject to these conditions. But if she wants intimacy and he does not want to do that, or there is the fear that she may do something wrong whilst she is still married to him, and the like, then he should divorce her, but still continue to spend on his children. And Allaah knows best.
With regard to the second question: the ‘iddah of a divorced
woman who has periods, with whom the marriage has been consummated and who
is not pregnant, is a matter concerning which classical and contemporary
scholars have differed. The view which is regarded as most correct by the
majority of contemporary scholars, such as Shaykh Ibn Baaz and Ibn
‘Uthaymeen, and others, is that the ‘iddah ends when three menstrual periods
have passed, and as soon as the third period ends, the ‘iddah ends. This is
the view of many of the major Sahaabah such as ‘Umar ibn al-Khattaab, ‘Ali
ibn Abi Taalib and Ibn Mas’ood, and it was narrated by Ibn al-Qayyim from
Abu Bakr, Abu Moosa and others (may Allaah be pleased with them all). (See
Fataawa Islamiyyah, 3/310; Fataawa al-Talaaq by Shaykh
Ibn Baaz, 193; Jaami’ Ahkaam al-Nisa’, 4/243).
With regard to the third question: “The women whose divorce is revocable (i.e., first or second talaaq) may uncover in front of her husband and adorn herself and wear make-up and perfume. She may speak to him and he may speak to her; she may sit with him and do anything with him apart from intercourse and the things that lead to it; that may only happen when he takes her back. (Shaykh Ibn ‘Uthaymeen, Fataawa Islamiyyah, 3/310)
If he kisses and embraces his wife with the intention of taking her back, then that taking back is valid, with no difference of opinion among the scholars. But if he does not intend to take her back, then some scholars say that it is permissible on the grounds that she is his wife, but that it does not mean that he has taken her back; other scholars say that embracing and kissing etc. are precursors to intercourse, so the one who does them is sinning if he does not intend to take her back. To be on the safe side, he should not do that until after he has clearly stated that he is taking her back, such as saying to his wife, “I am taking you back,” and two Muslim witnesses bear witness to his taking her back by him saying in front of them, “I ask you to bear witness that I am taking my wife So and so back,” and the like. Then he may do whatever he likes of permissible things. And Allaah knows best.
(See Subul al-Salaam, 2/267).