Schrödinger's cat thought experiment is meant to illustrate the idea of superposition, not entanglement.
Entanglement is a superposition that is shared between two different (usually thought of as spatially distant) parties. The mean different between a simple superposition is that when you talk of entanglement there are two parties that separate two parts of one shared quantum state.
You do not have this in the case of Schrödinger's cat, because in that case there is a single "observable" that is being measured (the "health state" of the cat).
Also, the statement
Due to quantum entanglement, if one electron (electron a) is spin up the other electron (electron b), which is entangled with electron a must be spin down."
is inaccurate. What is true is that two parties can share a specific type of entangled state of two spins which is such that whenever $A$ is measured to be, say, up, $B$ is measured to be down, and the other way around.
The "measured" bit is essential here: you cannot say that either electron is up or down before such measurement.