בתוך = Inside
לתוך = Into
Couldn't this also be translated "he RAN into the castle"? How would the Hebrew be different
Yes, both are correct. No way to tell the difference other than the context.
In fact running in should be בתוך not לתוך
That depends on whether the gate is open.
What is the difference between טירה and ארמון?
I think they are nearly the same. Maby טִירָה is a bit more of a "castle", but אַרְמוֹן more a "palace"