Two questions about astable multivibrators and transformers
But then if I put that output to a voltage doubler, the voltage should be 18V, not 9V right?


Correct.  Voltage doublers are easy enough that if you sorta-kinda understand how a capacitor and a diode works you can work this out for yourself, on paper.  Nothing does a better job of understanding circuits than working it out yourself.


  ...but something starts to smell burnt and the components of the astable multivibrator get hot...



transformers are inductive components, and inductors "want" to have zero average voltage on their terminals.  When you put your 0V/9V power to the thing, it sees a -4.5V/+4.5V signal added to a 4.5V signal.  It steps up the square wave, but it tries to short the 4.5V signal to ground.  That's going to cause an inordinate amount of DC current to flow.  The easiest way to deal with this is to put a capacitor between ground and the transformer -- it'll charge up to 4.5V, and everything will be fine in that regard.  Make sure your cap is big enough so that there isn't significant ripple, and that it can handle the current.
Your transformer is rated for 1/2 an amp at 9VAC.  A 555 can possibly drive that much, but it'll get hot.  Even if your transformer is unloaded, just the magnetizing inductance of the transformer is going to require a hefty current.  You really need a power stage there, that can safely drive an inductive load.