Is it a condition to verbally utter dhikr, recite the Qur'an, say "Bismillah," and perform all general remembrances, or is it permissible to suffice with doing so inwardly without verbal utterance?

For verbal adhkar (remembrances), it is necessary to move the tongue, and the majority of scholars hold that one must also make oneself hear it. Merely passing the remembrance or verses of the Quran through the heart without vocalization and moving the tongue is not considered a verbal remembrance, and it is not sufficient for the obligatory remembrances within prayer. As for outside of prayer, remembrance by the heart alone is an act of worship for which one is rewarded, but it is not on the same level as remembrance by the tongue along with the heart, nor does it encompass the virtue and reward associated with verbal remembrance. As for the verse: {And conceal your speech or publicize it; indeed, He is Knowing of that within the breasts} (Quran 67:13), it has no relation to this matter. The verse explains that secrecy and publicity are equal with regard to Allah's knowledge.