All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger.
Children are of two kinds: the first kind are the children who do not play in the mosque and who do not disturb the congregation; it is permissible to bring them to the mosque. The second kind are the children who play in the mosque and disturb the congregation, and they do not stop if they are told to stop; it is not permissible to bring these children to the mosque because their presence disturbs the congregation. For more benefit, please refer to Fatwa 88568.
Moreover, it is permissible to order the child to keep quiet and the like during the Friday speech but this should be done by pointing to him and not by talking to him; because it is not permissible to speak during the Friday speech as the Prophet forbade it.
If a person catches only one Rak’ah of the Friday prayer, he is considered as catching the Friday prayer and he should not pray Thuhr prayer instead. Not listening to the Friday speech has nothing to do with the validity of the Friday prayer as listening to it is not a condition for catching the Friday prayer.
Finally, it should be mentioned that it is disliked to be preoccupied during the speech. Therefore, that woman should not clean the mosque while the Imaam is delivering the speech, or do anything else that negates being submissive and listening to the speech.
Allaah Knows best.