I don't understand why "бежать в парк" means "to run TO the park" and not "to run IN the park". I would guess that the accusative mode used here is the reason for it, right? Then "run IN the park" would be "бежать в парке"? Even so there is still an open question: why "to run TO the park" is not translated as "бежать К парк"?
You're spot on about the usage of accusative vs prepositional. В+accusative refers to motion directed from outside into something; whilst в+prepositional refers to motion/state occurring wholly inside something.
As to translating 'run to the park' as "бежать к парку" (note that the preposition "к" takes a dative), it actually can be translated that way. But you have to remember the slight difference in meaning:
в+accusative refers to motion that leaves you inside the thing (think of the English 'into');
к+dative refers to motion that leaves you near the thing, but not inside (think of the English 'up to').
Like in english. "To the park" ( в парк) means you have the intention to be IN the park in the end. While "towards the park" (к парку) puts the intention only on the movement rather than the final purpose of being in the park. That's what i understand at least. I'm neither english nor russian native speaker though.
Yes, the accusative of an inanimate masculine object is similar to the nominative. This table helps (even though it is not exhaustive for the genetive): https://en.wikibooks.org/wiki/Russian/Grammar/Cases#Summary_of_declinations