Hello,
Thank you for posting your queries, answers to which are as follows:.
1. Yes, your cousin can very well re-marry and have a good life. If her future husband accepts her with the kids, then the law does not create any embargo against it.
2. After she obtains the divorce from her present husband, she can immediately re-marry and again, the law does not restrict this by any number of days or months thereafter as a breathing period. Thus, she can re-marry on the very next day as well.
3. The children's name does not automatically get changed after she re-marries. In case she wants the surname of her new husband for her kids, then the correct procedure is for the new husband to adopt the children. Thereby, they lose all the rights and claims to the property of their biological father. In order for adoption also, the permission of the biological father is required. However, if she wants to continue the old name for the children, then there is no need to go for adoption procedure. The biological father, can however, claim for guardianship of the children, so it is better to reach a mutual consensus on the issue of guardianship as well.
Please let me know if you have any further queries.
Regards,
Nishant