Why can 地 sometimes be omitted when placed after adjectives?
Is this because 大声 is originally an adverb (loudly) instead of an adjective (loud)?

Yes, you are correct.

In that case, if a word can act as either an ADJ or ADV, should I generally drop the 地?

If the word is primarily an adverb, you can drop the 地. For example, words composed with the typical adverbial suffix 然 don't require 地:

我们欣然欢迎朱大军同事加入我们公司 We cheerfully welcome the colleague Zhu Dajun to join our company

We can examine some cases where the adverbial phrase must have the verbal modifier 地. In these cases, the modifier is indeed an adjective:

他很清楚地阐述了那个故事 He told that story very clearly (清楚 clear -&gt; 清楚地 clearly)
我娘每天开开心心地做饭 My mum gladly/happily cooks every day (开心 happy -&gt; 开心地 happily)
飞机安全地到达了目的地 The plane safely arrived at destination (安全 safe -&gt; 安全地 safely)

Other phrases that are not exactly adjectives, but also not inherently adverbs, still require 地 to form an adverbial phrase, e.g. chengyu's:

学生们都翘足引领地看着考试结果 All the students are standing on tiptoes and craning their necks looking at the exam results
大楼鳞次栉比地挤在市中心 The big buildings are squeezed in the city center row upon row