"还早着呢。" How should we analyse this?
还: 仍然。
早: 时间靠前。
着: 音zhe, 轻声,助词,表示程度深。例: 好着呢!
呢: 助词,表示确定的语气。例: 好着呢!
You can understand the phrase as "Still very early" and "着" as "very" or "quite".
Your original understanding of "着" as indicative of "something in progress; the present continuous tense" is not correct. Neither is your understanding of 呢 as the same. 着 and 呢 have those functions sometimes, but not in this instance.
My personal preference is not to plug it into the English grammar of "subject + verb + adverb + adjective", for the very simple and obvious reason that Chinese is not English! Many Chinese "sentences" have no subject and no verb, and even the concept of what constitutes "a sentence" (一句) is vastly different between the two languages. Therefore Chinese should not be analysed according to English grammar. There is no 助词 like 吧 and 呢 in English, and I think it is counter-productive to classify them as adverbs.
My personal view is that it is one of the biggest fallacies in Chinese studies to apply English grammar to the Chinese language. I firmly believe that a foreign learner should approach the language like a pristine unblemished page (白纸一张) and enter into the Chinese language ethos like an infant, leaving his native language and English behind.
在北京大学对外汉语学院就有一个课程完全不用英语或者任何外语来教授汉语的,这是最好的,有一些国家的汉语精英都是这样培训出来的。说白了,就是让自己回到童年时学母语的状态。
中国同学学习英语的时候也可以参考这种方法。