What should a woman who delayed making up for missed fast, due to menstruation, till the start of the next Ramadan?
Whosoever broke fasting in Ramadan, and was able to make up for it, but didn`t until the next Ramadan started, is obliged to make up for the missed days, and to pay the ransom as well. However, if he/she wasn`t able to before next Ramadhaan due to an excuse, then he/she should only make up for the missed days. And Allah Knows Best.
Is the son of the sister by breastfeeding considered a mahram in Islam, and is it permissible for a woman to go for Hajj with him?
Everything that is prohibited by lineage is also prohibited by breastfeeding. The son of the sister by breastfeeding is considered a mahram, just like the son of the sister by blood, and it is permissible for him to be her mahram for Hajj and Umrah. And Allah Knows Best.
Has Islam made it incumbent on us to give our children particular names?
No, it hasn`t, but it is desirable that we give them good names.
Does Istigfar (Asking Allah for forgiveness) between the first and the second part of Friday prayer`s sermon render the prayer invalid?
Talking during Friday prayer`s sermon is disliked, but there is no harm in offering Istigfar between the two parts of the sermon, and it doesn`t invalidate the prayer as it is actually from Sunnah. And Allah Knows Best.