First, on all of your examples, you should prefix task with the, unless you mean that you're going to swap a specific task in for the literal word task in the sentence.
In addition, I'd consolidate your options into two, just for clarity and grammatical correctness. I've effectively merged your sentences and applied some grammatical sugar on top. These are prime candidates for comparison, which is your end goal.
- The task doesn't seem to have been completed.
- The task seems to have not been completed.
Your first two sentences and my first example sentence both put seem after does not. Seem in this context is much more about self doubt. It implies that you're not sure that it has been completed, and you're pointing out the potential incompleteness. The doubt is placed later in the sentence, which means that you yourself are doubting, not that you already know with confidence what has happened.
This is in contrast to sentences three and four of your example, as well as my second sentence. The seems comes first, which doesn't convey self doubt it at all. Instead, it carries the tone of suspicion. It says to the reader that you're very aware that the task hasn't been completed, but you want to be slightly tactful and not outright say that it wasn't done. In a more general sense, seem is often used rhetorically to fluff up a fact so that you can sound humble, but still proudly be correct. The key is the early placement of the word in this context.