Pep boys has gone belly up everywhere but down there...
and the free diagnostic by pimple faced teenage boys is worth what you paid for it.
they put a USED alternator on it? what is the warranty?
I think you shoulda spent $20 on a voltage meter, and figured out what the real problem is.
do you have a test light? ($5)
I can walk you through diagnosing it if you can find a meter...
i'd lean toward the starter with the current information, but it won't cause it to die.
push starting your corrolla won't hurt it unless you do something weird and crazy.