After the expiration date of the marriage...
After the expiration date of the marriage, the couple must either extend it to another specified time or they will automatically separate without divorce, and the woman must count the "Iddah" (waiting period) before which she can not marry any one else. The dowry (Mahr/Sidaq) should be specified at the time of contract. It may be a calculated sum of money, or a certain work like teaching a certain science, or something like a ring or bracelet by specifying or showing them.
As for the dower which are not present at the time of contract, it is sufficient that the dower be described in such a manner that the woman's misunderstanding will be removed, i.e., that it be described exactly as it is. Usually in this type of marriage, the couple do not plan to have any child. But it becomes accidentally the case, then the kids from this marriage are the legal sons/daughters of their parents. All regulations of permanent marriage will apply to the children.
The man and woman are husband and wife till the expiration date of the contract, much in the same way as the normal marriage is.
In fact the author of al-Jawahir (the biggest Shia Encyclopedia in Jurisprudence) said: "Originally the same rules apply to permanent and temporary marriage, except what was excluded by certain proofs." For instance, similar to the permanent one, in the temporary marriage it is necessary for the virgin girl to get the permission of her guardian (which is usually her father) before accepting such offer.
Imam al-Ridha said: "A virgin may not be married temporarily without her father's permission." (Wasa'il al-Shia, v14, p458). As for a non-virgin, she can make her decision alone without getting the agreement of her guardian or any other person, but it is still encouraged to get the permission. Temporary marriage is valid with the Muslim women and the woman of the People of the Book like the Christians and the Jews.
And if the Christian/ Jewish girl is virgin, the prosper should still seek the permission of her guardian even if that guardian is not a Muslim. Under very extraordinary circumstances, if it is proved that her guardian does not care about her benefit, a Muslim scholar (or the judge when the government is truly Islamic) can let her marry. In this case the necessity of the permission of the father is lifted and will be shifted to the permission of the scholar/judge.
This is true for permanent marriage as well.