1st case: Definitely a carrots and sticks approach. Demonstrating the moral or material superiority of your religion can't hurt. Having dedicated missionaries willing to live amongst the locals and understand their culture will aid conversions.
And then the sticks: A special tax for non-believers seems to be an effective long-term tactic, though it may not eliminate opposition religion entirely.
Destroying the local clergy/ houses of worship can't hurt either.
2nd: Hard to say. There are very few areas where Islam took root that it has been supplanted or disappeared except for Spain, and that took the expulsion of the population. For sheer longevity Judaism is hard to beat. Hinduism is rather resilient in its own backyard as well.