Do pracy jadę metrem, bo metro jest szybsze niż autobus.

Because do is the normal preposition for movement to a destination, and it takes the genitive case.

• praca = work
• do pracy = to work

So iść do pracy / jechać do pracy means to go to work.

Na pracę does not mean to work in this sense. It could appear in other contexts, but not as the normal expression for going to your workplace.

Jadę comes from jechać, which means to go / travel by vehicle.

So:

• idę = I go on foot
• jadę = I go by some form of transport

Since the speaker is using the metro, jadę makes sense.

A useful nuance: if you are talking about a habitual repeated commute, many speakers would very naturally say Do pracy jeżdżę metrem.
So:

• jadę = I’m going / I go by vehicle, often one trip or a present situation
• jeżdżę = I go regularly / habitually by vehicle

Textbook sentences sometimes use jadę in a broad present-tense way, but in everyday Polish jeżdżę is often more natural for a regular commute.

Because Polish often uses the instrumental case to express the means of transport.

So:

• jadę autobusem = I go by bus
• jadę pociągiem = I go by train
• jadę samochodem = I go by car
• jadę metrem = I go by metro

Here, metro changes to the instrumental form metrem.

Because in the second clause, metro is the subject of the sentence:

metro jest szybsze niż autobus
= the metro is faster than the bus

Subjects normally appear in the nominative case, so the form is metro.

So the two forms have different jobs:

• metrem = by metro, means of transport
• metro = the metro, the subject of jest

Because adjectives in Polish must agree with the noun in gender, number, and case.

The noun metro is neuter singular, so the adjective must also be neuter singular:

• masculine: szybszy
• feminine: szybsza
• neuter: szybsze

So:

• autobus jest szybszy
• kolej jest szybsza
• metro jest szybsze

Because szybsze is an adjective, and szybciej is an adverb.

Here the sentence compares two things:

• the metro
• the bus

So Polish uses an adjective:

• Metro jest szybsze niż autobus.
  = The metro is faster than the bus.

If you were describing how something moves, you might use the adverb:

• Metro jeździ szybciej niż autobus.
  = The metro travels faster than the bus.

Both are possible in Polish, but they are structured differently.

Because after niż you are comparing the metro with the bus as a thing, not saying by bus.

So:

• metrem = by metro
• autobusem = by bus
• autobus = bus, as the noun being compared

In metro jest szybsze niż autobus, the word autobus is the thing being compared with metro, so autobus appears in its basic form here.

You may also see another common comparison pattern:

• Metro jest szybsze od autobusu.

That uses od + genitive:

• autobus → autobusu

Both mean the same thing.

Because Polish usually drops subject pronouns when they are clear from the verb ending.

The ending -ę in jadę already tells you the subject is I.

So:

• jadę = I go / I’m going

Adding ja is possible, but it usually adds emphasis or contrast:

• Ja jadę metrem, a on autobusem.
  = I go by metro, and he goes by bus.

Without special emphasis, Polish normally omits ja.

Yes, ponieważ is possible.

• bo = because, everyday and very common
• ponieważ = because / since, a bit more formal or written

So both work:

• Do pracy jadę metrem, bo metro jest szybsze niż autobus.
• Do pracy jadę metrem, ponieważ metro jest szybsze niż autobus.

For normal conversation, bo is very natural.

No. Polish word order is fairly flexible, because case endings show how words function.

This sentence can be rearranged in ways that are still grammatical, for example:

• Jadę do pracy metrem, bo metro jest szybsze niż autobus.
• Metrem jadę do pracy, bo metro jest szybsze niż autobus.

The version Do pracy jadę metrem... puts to work first, which helps set the scene or topic.

So the order can change, but the emphasis changes too.

Yes, you could say that:

• Do pracy jadę metrem, bo jest szybsze niż autobus.

That is understandable, and jest szybsze would naturally refer back to metro.

But repeating metro makes the sentence clearer and more explicit, especially in teaching materials.

Using ono would be less natural here unless you wanted special emphasis:

• bo ono jest szybsze... sounds emphatic

So in normal Polish:

• repeating the noun is fine
• dropping it is also fine if the reference is obvious
• using the pronoun is less common in this kind of sentence

Polish does not have articles like a and the.

So:

• metro can mean the metro or a metro
• autobus can mean the bus or a bus

The exact meaning depends on context.

That is why Polish learners coming from English often feel something is missing, but in Polish nothing is missing at all.