Is it permissible to announce the commencement of prayer via loudspeaker, or is that exclusively for those present inside the mosque? And is there anything in the Sunnah that prohibits raising one's voice with it?

There is no harm in announcing the call to prayer (Iqama) via loudspeaker. The Sunnah indicates its permissibility, as the Iqama was audible from outside the mosque during the time of the Messenger of Allah, peace and blessings be upon him. This can be understood from Abu Hurayrah's narration: "When you hear the Iqama, walk to the prayer with tranquility," and from Ibn 'Umar's narration: "When we heard that the prayer had been established, we would perform ablution and then go out for the prayer." Some scholars, such as Shaykh al-Islam Ibn Taymiyyah and Ibn Qudamah, mentioned that the Iqama is an announcement for those present and absent from the mosque to stand for prayer. Shaykh Ibn 'Uthaymeen supported this with the aforementioned Hadith of Abu Hurayrah.