I firstly confirm that I am not a native speaker. However I would share my feeling when reading your sentences. To me, yours seem natural.
I, TS. Lę V, not an English teacher.
Student or Learner
Are the following sentences natural to a native ear?
1. I was wondering where you are headed and do you mind if I ride out with you?
2. I heard that line in a movie I was watching yesterday.
3. That is why I was trying to see if finding combinations of these factors would help.
The first one is not natural. I would say
"I was wondering where you were headed and if you wouldn't mind giving me a ride".
I wouldn't use "ride out" as this has a slightly different meaning. You "ride out" something difficult like a storm or an economic crisis. You can use it to "ride out of town" but this really depends on where you are and what the locals say about the point you are going to.
Note: "ride" is US English. In British English it's "lift"