Does an internal examination require ghusl?

Ghusl isn't required for making internal examination; however, it invalidates one's fasting. And Allah Knows Best.

Is it permissible for a woman whose menses exceeded (15) days to pray, fast and do what was prohibited on her to do during the latter period?

If the menses exceeded 15 days then it is deemed menstruation, and in this case she is obliged to pray and fast and to do what was impermissible for her to do during the latter period. In addition, she is obliged to make an ablution for each obligatory prayer after it's due time and making sure that impuity is removed. Finally, your menstruation period is over (Questioner) simlar to your menstruation in the previos month.

What`s the ruling on using the following materials in manufacturing cosmetics: olive oil, Vaseline, talc powder, glycerin, honey, almond oil, pine oil, galingale and marjoram?

All perfect praise be to Allah, The Lord of The Worlds, and may His peace and blessings be upon our Prophet Mohammad and upon all his family and companions.                                                                                                                                                                                                                                    None of the above materials is forbidden, and it`s lawful to use them in manufacturing products that are beneficial to people. This is because, in principle, all things are lawful so long as there is no evidence in Sharia indicating otherwise. And Allah Knows Best.

Is the woman who do breastfeeding permitted to break the fast during Ramadan?