What is the ruling on saying the Takbir, reciting the Quran, and making supplications in prayer with a voice that is between audible and inaudible? Is it permissible to raise one's voice slightly so that people can hear, or is it sufficient to recite only to oneself?

It is sunnah for the Imam to recite aloud and to pronounce the Takbir and Tasmi' aloud so that the congregants may hear. It is sunnah for the one praying alone and for the congregant to recite these silently. The minimum for silent recitation is for the worshipper to hear himself. Both loud recitation in its proper place and silent recitation in its proper place are sunnah, and neglecting either does not invalidate the prayer. As for supplications (du'as), it is sunnah to recite them silently in all cases (for the Imam, the congregant, and the one praying alone), and the prayer is not invalidated by reciting them aloud.