I make a cake. I am a cause of the cake. The cake contains eggs. I am not the cause of the eggs. I think "what causes the whole, causes each part" is a bad intuition to have.
In general, I think it is an error, and a source of confusion, to think of things rather than events having, or being, causes. I know people sometimes do, and I've gone along with it in #1 above, but I think it's a mistake.
Why would anyone assume A3? It seems really arbitrary. Exception: you might believe A3 because you believe in an entity of which all others are parts. See below.
If E includes an entity V of which all others are parts (call it "the universe") then, provided C is reflexive, V C* anything-you-like. And I think it'll then turn out that the way all your theorems work is that V is the canonical uncaused cause of everything. Which is a bit dull and wouldn't satisfy many theists. Perhaps something more interesting happens if you make C irreflexive instead, so that things don't count as causes of themselves.
Fair points, though there is in fact a lot of disagreement about what are the basic relata of the causal relation: see the SEP entry for example. When we apply causation to entities (which we can sometimes do, as in your example) then "A causes B" probably means something like "at least one event in which A is involved is a cause of every event in which B is involved".
On counterexamples to "what causes the whole, causes the part" : possibly an even stronger counterexample considers just one of the atoms in the cake. However, ...
If it's worth saying, but not worth its own post (even in Discussion), then it goes here.