Is it permissible for a believer to avoid researching and reading about other religions for fear of apostasy, or is that considered a doubt in their faith?

This fear is legitimate and does not stem from doubt in religion. Rather, it is a fear of some ambiguities remaining in the heart. For this reason, the Imams of the Salaf warned against listening to and spreading ambiguities, because hearts are weak and ambiguities are captivating. It is not permissible for ordinary Muslims to be exposed to the ambiguities of other religions, for fear of confusing them. This is part of preserving the religion, not a sign of doubt. Al-Lalaka'i, in his Sharh Usul I'tiqad Ahl al-Sunnah wal-Jama'ah, dedicated a chapter titled "Narration of what was reported from the Prophet, peace and blessings be upon him, concerning the prohibition of debating with people of innovation, arguing with them, conversing with them, and listening to their newly invented sayings and evil opinions."