The sentence is correct. It emphasizes the priority of one action over the other - that is why two different past tenses are used. Don't be confused by inversion - change of standard word order, it serves to stress the priority. On the whole, it is a typical grammatical pattern, a bit bookish perhaps.

The word that should be excluded. It leaves the second part of the sentence without a predicate.

It means that it may be one way or another, there are too many factors that affect it. You say so, if you do not want to say yes or no. Sometimes it's a phrase to evade a direct answer.

No, you should say: I looked him straight in the eyes.

Both sentences are gramatically correct, but the second one sounds a bit more emotional and imperatative (like - if you haven't yet seen the doctor, you should certainly do it ASAP!).