1. Yes, this is correct but normally you would say "A stranger.." Yes, you can omit "the" before "directions" but it is slightly different. You would use "the" if you want to emphasize "directions" in the sentence. For example, if you are explaining you got lost because the directions were incorrect and "a stranger gave you the directions". 2. Yes, this is correct

The second sentence is correct. Well done!

Yes, you can omit "the". That is because in this case, "the directions" can refer to the set of directions, where as simply "directions" is the plural of direction, and refers to the specific steps he gave you in order to get there. E.G. "Turn left" is one direction, where as "Turn left, then turn right." are directions. I'm actually not sure about the seconds one, though. Sorry.