По дороге в метро я повторяю новые слова.

In по дороге (meaning on the way / along the road), по takes the dative case.

• дорога (dictionary form, nominative) → дороге (dative singular)
  This whole phrase works like an adverbial expression: по дороге = while on the way.

по дороге в метро is an idiomatic way to say on the way to the metro.

• в + accusative often marks a destination (going to a place as an endpoint).
• к + dative means towards / up to something (often “to” a person or “to/near” a place).
  So both can be possible depending on nuance:
• по дороге в метро = on the way to the metro (as a destination)
• по дороге к метро = on the way towards the metro / up to the metro entrance

Yes, Russian word order is flexible. Putting По дороге в метро first sets the scene (time/situation) and sounds natural. Common alternatives:

• По дороге в метро я повторяю новые слова. (neutral, sets context first)
• Я по дороге в метро повторяю новые слова. (focus a bit more on я)
• Я повторяю новые слова по дороге в метро. (focus more on the action first)

You can omit я if it’s clear who the subject is:

• По дороге в метро повторяю новые слова.
  Russian verb endings show the subject, so повторяю already implies I. Keeping я can add emphasis or contrast.

повторять (imperfective) fits a habitual / repeated / ongoing meaning: you do it regularly on the way.
повторю (perfective, from повторить) would mean a single completed repetition, often with a “result” feel:

• По дороге в метро я повторю новые слова. = I’ll review the new words (and finish that review) on the way.

It’s the direct object of повторяю, so it takes the accusative.
Because слова is inanimate plural, accusative plural = nominative plural in form:

• nominative plural: слова
• accusative plural (inanimate): слова
  And the adjective agrees:
• новые (plural, matches слова)

Yes. For animate plural nouns, accusative plural usually matches genitive plural.
Example pattern:

• inanimate: вижу столы (same as nominative plural)
• animate: вижу студентов (matches genitive plural студентов)
  In your sentence, слова is inanimate, so it stays слова.

A very common alternative is по пути:

• По пути в метро я повторяю новые слова.
  по дороге feels a bit more “literally on the road/route,” while по пути is often more general (“along the way”), but in many contexts they’re interchangeable.

Typical stress:

• по дорогЕ
• повторЯю
• нОвые
• словА
  Sound change: в метро is often pronounced with devoicing/assimilation, close to ф метрó in fast speech.

You can, but it’s different in style and structure:

• по пути = a common conversational adverbial phrase (along the way)
• в пути = more like while en route / in transit, often used with verbs like быть:
  
  • Я в пути. = I’m on my way / I’m en route.
    So for your sentence, по пути в метро is the more natural match.