There is a semantic difference between look/appear and seem:
One should use look or appear when describing an observable condition - e.g.: Rhonda looks sad - the example implies that there is some observable state or behavior that supports the statement (Rhonda may have tears rolling down her cheeks, for example).
One should use seem to describe a perceived condition - e.g.: Rhonda seems sad - the example implies that the person making the statement has a perception regarding the condition of the subject.
You're correct, it is indeed contradictory. Taken purely logically, then, if neither can live while the other survives, then if Harry is alive Voldemort must be dead and if Voldemort is alive Harry must be dead. Since we know both Harry and Voldemort are alive, the statement is clearly false. Since the statement is part of a piece that refers to them as alive, and one killing the other, then the passage as a whole is clearly illogical.
However, there are a few sensible ways to read it.
One is to suggest that their life under this doom (in both the common sense, and the original sense of being fated to something) is incomplete. Neither truly live until after it comes to pass.
The other, is that as a prophesy it is speaking about the future, so even though the present tense is used, it should be thought about in terms of the future. This is an unusual use of tenses for most cases, but reasonable if we consider that we're talking about a magical trance that leads people to speak dialogue in a completely different register and rhythm than that the author normally uses! And this also makes it make perfect sense; some point in the future will come, in which one of them must be dead in order for the other one to survive.
Spoiler: Hover over the text to read it if you have read the end of the last book, or don't care about the plot being given away:
It also has a different interpretation, that becomes clear later. Voldemort is in a state that is neither life nor death, and for that reason cannot be killed. This state can only be ended when certain objects are destroyed, and Harry is one of those objects. This means that while Harry survives, Voldemort cannot truly live, and cannot truly die. Harry dies, and comes back to life. Harry's death is the destruction of the last object that keeps Voldemort in his non-life/non-death state, so Voldemort truly lives when Harry stops surviving. Then they can kill him.
The strangeness is justified as fitting with the general strangeness and cryptic opacity of the prophesy as a whole (again, especially in taking how it doesn't fit with the normal style of the books). It's justified further in light of the plot twist above.
It's also deliberately strange in its phrasing, because you're meant to be wondering about it until you've read on through the series.
Best Answer
Logically speaking, the verb conjoin really should be acceptable. A conjunction is the act or product of conjoining. It's the same stem, so if one form is deemed sufficiently precise to refer to the operation, why shouldn't the other be? The counterpart, for "or-ing", would be disjoin. Conjunction has a more specific meaning in propositional logic than in general and grammatical usage. Under the logic that two statements could be "conjoined" with the OR operator, all disjunctions would also be conjunctions. That's obviously not a very useful sense in context.
To clarify to readers the fact that you're using the word in a context-specific way, the first time you use this verb, simply follow it with "(i.e. join with AND)" or "(as opposed to disjoin)". If you're going to use the verb that way repeatedly in a longer text, you might briefly but explicitly explain in a footnote instead. Given how infrequently we use conjoin in general language, I don't think this overloading of terminology would be likely to cause ongoing confusion. (It's not as if it will be mixed with uses of that verb in its everyday sense.)
Personally, I think "conjuncted" just sounds influent.