I would not call them synonyms, though both involve rapid movement, in their intransitive senses:
'Bolt' has an underlying sense of 'sudden swift movement by an animal, after a fright': i.e. sudden movement away from something (cf. 'cheval emballé).
'Dart' has an underlying sense of 'sudden swift movement (for any reason) towards something
'Bolt' seems to derive from a Germanic root ('bolz'). There is an ancient Greek verb 'ballein', which relates to 'throwing missiles' (cf. 'ballista'); but I don't know if the two words are related.
Unfortunately I don't have a French etymological dictionary to hand, so I don't know whether there's a connection between 'emballer' a
Morning (is it?), Mr P. Well I've just checked, and "emballer" & "bolt" seem to have the same root: In French, "balle" meaning a charge, or some kind of luggage and "balle" either to shoot or to play with come from the same word, obviously derived from ballein. As to "emballer, or s'emballer", you just add um- > em-. So, "emballer" would mean make a bundle that looks like a ball
Yes, you can; though that does imply that after a sudden fright, 'she' (a cat, perhaps) shot towards the door and – at the door – suddenly stopped. (Perhaps there was a dog outside.)
'Bolted for the door' is more usual, and would imply that she probably didn't stop when she reached the door.
On the other hand, both phrases are quite close to 'she bolted the door', as in