Ayşegül, "this" var olduğunda hiç bir "the" var olamaz. "from this palace" (bu saraydan) veya "from the palace" (saraydan). :-)
Right: "bu" is functioning as an adjective, here, and adjectives don't decline with their nouns.
is this used when talking about your nationality, like i come from the USA
Yes :) "Ben Amerika'dan geliyorum" but even better would be "Ben Amerikalıyım"
Small correction to your English though...you would have to say "I am from the US." or even better "I am from the US." :)
Why not 'leaving this palace'? Actually I can guess why - that would be 'Ben bu saraydan gidiyorum' (I am going from this palace.) In English 'I am coming from this palace would be said when at a distance from the palace looking back. I am leaving this palace would be said when at the palace. I guess it's the same in Turkish?