No, it's omitted for a different reason. "Most furiously" is a rather stronger version of "very furiously"; from a grammatical standpoint the two words can be used interchangeably in this context. "Deepest" doesn't have the article simply because it is often optional.
In this case, this lake is at its deepest at this point. The "at its" is implied. It's hard to make a construction with "the deepest" in it that has the same meaning. "This lake is the deepest lake at this point" implies comparison to other lakes, not other points in the same lake. "This lake has the deepest point at this point" not only sounds awkward, but "has its deepest point" is more typical of English than using "the".
These three sentences come to mind which may help:
This thingamajig is the most spectacular of the year.
This thingamajig is most spectacular in the early evening.
This thingamajig is most spectacular.
In the first, it is the thingamajig that is spectacular, so it is the most spectacular thingamajig and we just don't want to say thingamajig twice. In the second, "at its most spectacular" is implied. In the third, we are simply saying that it is rather more than very spectacular.
Edit: I wrote this before you specified exactly where the "the" went in each sentence, and looking at your change gave me another idea to share. When you say "the war rages the most furiously in this region" it means that this region is where the war is most furious, as compared to other regions. When you leave out the the, it has the meaning of more than very furiously. So, in this case, interestingly, it can completely change the meaning of the sentence. (You can use word emphasis to keep the meaning the same, however. Roughly, if you say the war rages MOST furiously in this region, it means very furiously. If you say the war rages most FURiously in this region, it means the same as the most furiously.)