Is it permissible to request financial assistance from charities or individuals to support an Islamic website that hosts Quran recitations, for the purpose of renewing its hosting, given the fear of engaging in "prohibited begging," and is it permissible to place paid advertisements for other websites on this site to fund it?

There is no legal impediment to seeking assistance from NGOs and charitable organizations to support a website that broadcasts the Quran, because this is a public interest that serves religion, and it is not considered a blameworthy request as long as it is not for personal gain. The Prophet, peace and blessings be upon him, urged people to give charity to help the needy, as stated in Sahih Muslim, and "Whoever introduces a good practice will have its reward and the reward of those who act upon it." The blameworthy request is for a person to ask for money for himself to increase his wealth, as explained by Al-Manawi. A Muslim must seek help from Allah while taking the necessary means.