Learn Hardcore Russian: На узкой дороге обгонять опасно, потому что другая машина может появиться неожиданно. - On a narrow road overtaking is dangerous because another car can appear unexpectedly.

Because на + Prepositional case is commonly used to mean on / along (a surface or area) in the sense of location: на дороге = on the road.
So дорога → (на) дороге is Prepositional singular.

Why does the adjective become узкой?

Adjectives agree with the noun in gender, number, and case.
дорога is feminine singular, and after на (location) the noun is Prepositional, so:

узкая дорога (Nom.) → на узкой дороге (Prep.)

What’s going on with обгонять опасно? Why an infinitive?

Russian often uses the pattern Infinitive + (category of state) to express general statements like It’s dangerous to…
Here обгонять = to overtake, and опасно is a predicative word meaning dangerous.
So обгонять опасно = (It is) dangerous to overtake.

What part of speech is опасно here?

Here опасно functions as a predicative / category of state (similar to нужно, можно, нельзя, трудно). It’s not describing a noun; it’s stating a condition about the whole situation: (it is) dangerous.

Why isn’t there a word for it (like это) or a verb like is?

In the present tense, Russian normally doesn’t use a verb “to be” (есть) in sentences like this, and it also often doesn’t need a dummy “it”.
So instead of It is dangerous…, Russian simply says Обгонять опасно.

Why is there a comma before потому что?

Because потому что introduces a subordinate clause of reason (because…). In Russian, the main clause and the потому что-clause are normally separated by a comma:

…опасно, потому что…

Why другая машина and not другой машина?

Because машина is feminine, so the adjective must be feminine too:

masculine: другой
feminine: другая
neuter: другое
plural: другие

So другая машина = another car / a different car.

Does другая машина mean “the second car” or “another car”?

It means another / a different car, not specifically “the second one.”
If you meant “the second car” in an ordered sequence, you’d more likely use вторая машина.

Why is it может появиться (perfective), not может появляться?

появиться (perfective) focuses on a single event: a car may (suddenly) appear.
появляться (imperfective) would suggest repeated/ongoing occurrences or the process of appearing. In a warning about a sudden oncoming car, perfective появиться is the natural choice.

Is появиться reflexive because of -ся?

Not in the “someone does it to themselves” sense. Many Russian verbs with -ся are simply intransitive / “happen” style verbs.
появиться means to appear / to show up (something comes into view), and -ся is just part of how that verb is formed.

Where can неожиданно go in the sentence? Is the word order flexible?

It’s fairly flexible, but the placement affects emphasis. Common options:

…может появиться неожиданно (neutral: appear unexpectedly)
…может неожиданно появиться (more focus on the unexpectedness) Both are correct; Russian often moves adverbs for emphasis.

Could the first part also be На узкой дороге опасно обгонять?

Yes. Both are natural:

На узкой дороге обгонять опасно (focus slightly on overtaking)
На узкой дороге опасно обгонять (focus slightly on dangerous) The meaning stays essentially the same; it’s mostly about what you highlight.