Disproving Odin - An Experiment in arguing with a theist with Theist logic

[Jenny A]

Syllogisms cannot be used to prove the existence of things.  There's nothing wrong with the Socrates is mortal syllogism because it does not attempt to prove the existence of immortals.  

All men born of women are mortal
Socrates was not born of a women
Therefore Socrates is immortal

Anytime a syllogism provides new facts about the world instead of sorting out the facts we have, it involves a fallacy of some kind.

I'm not sure whether you are limiting yourself to the old Aristotelian syllogisms, but the more modern versions absolutely can prove the existence of things.

So, The statement 'there is an x such that P(x)' is equivalent to 'it is false that for all x, P(x) is false'. So, to prove an existence, we simply need to negate a universal claim. So,

All A are B.
C is not B
-----------
There exists something that is not A.

is a correct deductive form.

Your claim that syllogisms do not provide 'new' facts about the world is, while technically true, very misleading. As an example, take any non-trivial result in mathematics. That result is proven via syllogisms from previous results. Hence, in a sense, it is not 'new knowledge'. But that is clearly not a useful way to talk about things. The new, non-trivial, result is definitely new knowledge that was not clear from the previously known material and in which the syllogism produced a new perspective.

More specifically, suppose we start out in geometry class with the basic axioms for lines and angles. Later, the Pythagoren theorem is proven from those assumptions. To say that is not 'new knowledge' even though it is 'contained' in the axioms seems to be a very strange way to use the language.

I'll try to be more precise.  Syllogism can not be used to show the existence of things not contained in the premises:

All A is B.
C is not B
-----------
There exists something that is not A.

The conclusion is true, but it does not conclude that there are things other than As, B's, and Cs. All of the not-As proven in the conclusion are Cs.  Whether there are not-As which are also not-Cs is not shown because nothing in the premises contains information about things which are not A, B, or C.  

 What it does do is tell us something new about the relationship between As, B's, and Cs.  I'm not suggesting that new propositions about A, B, and C aren't useful.  They certainly can be.