Why can objects be placed at the beginning of the sentence? Often, I come across sentences like below, where the subject is missing and the object is placed instead.

证件带了吗？ （Instead of 你带了证件吗？)

I am wondering if there is a name to such grammatical structure, and if there are any conditions to be aware of when I want to use such structure. Thank you!
It's called topic-comment structure and it's ubiquitous in Chinese. It's one of its distinctive grammatical traits.
What is called topic, or theme, is the focus of the sentence and is extracted at the beginning of the sentence, regardless of its grammatical function. This last remark is especially important. The topic isn't necessarily also the grammatical subject of the sentence.
What is called comment, is  usually what contains the predicate and provides the additional information about the topic that you want to convey. In wikipedia's words &quot;it's what is being said about the topic&quot;.
When the topic is a complex structure, it's often separated from the rest by a comma in writing, and by a brief pause in speech. In writing the comma may disappear for simple topics. But you can add it again to test this:

证件，带了吗？

证件 is the topic, and the grammatical object of the sentence. The two are not mutually exclusive.
（你）带了吗 is the comment, and the grammatical predicate. The actual subject 你 is omitted, as it's understood from the context (supposedly this sentence is being addressed at you).

Other examples where the topic is the object (with and without comma):

饭吃了吗 = Have you eaten already?
别人的中文名字，你一定要记住啊 = Make sure to remember other people's Chinese names!