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.
Hu rats letokh ha-tira.
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"
He runs into the palace should be accepted
The ר in רץ is pronounced very pourly, at least its hard for me to hear it
Is he colliding with the castle or is he running through a gate? Could it be both?
Well, as תָּוֶךְ means the inside, interior, the usual interpretation should be that he entered the castle in order to be inside it, but you can use it to express a heavy collision לְתוֹךְ הַקִּיר *into the wall.