For example, if Peter is my source, should I say:
Peter mentioned his '... unquenchable thirst, a fatigued body...' as being part of the reason for his actions.
Or would I have to leave out the ellipses?
No, you would leave out the ellipses there.
The Purdue OWL has a page about this; it lists this example:
Even if you aren't quoting Peter's remarks in their entirety, you don't need to use elipses, because your sentence is structured in a way that shows you are only using a small segment of his overall quotation (much like the preceding example).
The same site goes on to say:
So, if Peter had said:
then you might say:
Going back to your example, there is another way you could structure the sentence, to avoid the awkwardness of the article a:
One generally does not place an ellipsis at the beginning of a quotation to indicate the omission of material, because it is usually evident (as in your example) that the quotation is only part of the original.
However you should use an ellipsis if the words as they appear in your quotation could be mistaken for a complete sentence, but in the original are only part of a longer sentence.
This page has a good example of where you should use ellipsis in the beginning of a quote:
In general, it is not necessary to put ellipses at the beginning or end of a quote. Any quote is an excerpt from a larger statement. We know you are not quoting everything that the person ever said or wrote in his life: it is not necessary to tell us that this person said other things before and after what you are quoting.
Personally, though -- and I can't give a reference on this, it is just my opinion -- I do include ellipses when I am leaving out potentially relevant material, even if it comes at the beginning or end.
For my purposes in the quote I may only care about these three, but if he listed others, it could be misleading to end the quote without some indication that the list continued.