Past Tense – Did Remember vs. Remembered

You are correct. Present tense indicates that the action occurs in the present. Past tense indicates that the action occurs in the past.

I remember his blue tie.

I remember (now) his blue tie.

I remembered his blue tie.

I remembered (at some point in the past) his blue tie.

When the police detective was questioning me yesterday about the bank robbery the day before, I remembered the robber's blue tie.

In the passage quoted, the author shifts into the past tense for reasons unexplained. A practiced author would be very aware of tense, but this sort of shift is not uncommon among ordinary speakers. When they are narrating a series of events, their grammatical choices often reflect an unspoken thought. Or it could be that the author of the summary is referring to Holly's seeing this man at the bar on a number of occasions in the past, and the "pastness" of those sightings caused the speaker to choose "remembered" instead of "remembers". Another possibility is that the present tense is merely a formal convention for such summaries. In reality, the author of the summary has read the book in the past, and is describing something from his or her own past (memory of the book's plot), and the natural tendency would be to use the past tense.

I have to go North

This is a common way of expressing obligation in modern English.

I am to go North.

This is a rare, possibly old-fashioned, or extra formal way to express future events that can't be avoided. Not so much obligation, but a certain future.

You are to be executed at dawn.
You have to be executed at dawn.

Comparing these. The second suggests that the speaker would try to stop the execution. The first doesn't carry that implication. A judge passing sentence would use the first expression.

People rarely talk like this, except when acting in films with an "old-world" theme.