They are not correct as written--well, they are actually grammatically correct--but it's just a minor syntax issue that misconstrues the meaning. As you've written them, they, in reality, mean something different from what you'd intended to say, which I'll explain in a bit.
As the commentators have suggested so far, it should read more like this:
That task could very well be
very well assigned to me
That task could well be
well assigned to me.
First of all, yes, you absolutely can omit very. It's just an intensifier, and intensifiers are only required for emphasis. When you remove very, you make the likelihood less certain. But it's still grammatically correct.
Second, in these examples, as you seem to be aware, well is the equivalent of definition number 10b under the adverbial definition. In this source, it's definition number 4b.
Now, as for the way these sentences were originally constructed, that would mean that it'd be a wise choice to assign the task to that person.
That task would [could] be [very] well (i.e., properly) assigned to me.
In the first set of examples, well means likely or possibly. In my last example, it means that it'd be a good decision to assign the task to that person.
All of these are examples of speculation in the subjunctive mood.