Is it permissible for a woman to leave her house wearing brightly colored clothing?
Brightly colored clothing is of woman's adornments, however; and this shouldn't be revealed save to her husband and Mahrams. And Allah Knows Best.
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 permissible to buy/sell with non-Muslim?
Yes buying/selling for a Muslim with non-Muslim is permissible as regards what is lawful.
What is the ruling on praying behind an undesirable Imam?
It is permissible to pray behind an Imam who is undesirable for committing religious violations, and it is disliked for such an Imam to lead the prayer. Still, If he was undesirable for a personal, or a worldly matter, then such argument doesn’t hold water. And Allah Knows Best.