Я жду такси уже полчаса.

Because Russian does not have a separate tense for I wait vs. I am waiting.

The verb ждать is imperfective, and its present-tense form я жду can mean:

• I wait
• I am waiting

So in this sentence, я жду naturally means an action that is happening right now.

The dictionary form is ждать = to wait.

Жду is the 1st person singular form, so:

• я жду — I wait / I am waiting
• ты ждёшь — you wait
• он/она ждёт — he/she waits
• мы ждём — we wait
• вы ждёте — you wait
• они ждут — they wait

So жду simply means I wait / I am waiting.

Because Russian ждать usually takes its object directly, without a preposition.

So Russian says:

• ждать такси — wait for a taxi
• ждать автобус — wait for a bus
• ждать друга — wait for a friend

English needs for, but Russian does not here.

It is the object of ждать.

The tricky part is that такси is an indeclinable noun, so its form stays такси instead of changing its ending. That means the form looks the same in different cases.

So for a learner, the important point is:

• такси is the thing being waited for
• its form does not change

This is very common with some borrowed nouns in Russian.

Уже literally means already, but in this kind of sentence it often sounds more like:

• already
• by now
• for ... already

So уже полчаса gives the idea that the waiting has lasted a noticeable amount of time.

Compare:

• Я жду такси полчаса. — I’ve been waiting for half an hour.
• Я жду такси уже полчаса. — I’ve already been waiting for half an hour.

The version with уже often suggests impatience or emphasis.

Полчаса means half an hour.

It is built from:

• пол- — half
• час — hour

So:

• полчаса — half an hour
• полдня — half a day
• полгода — half a year

This is a very common fixed pattern in Russian.

Russian often expresses duration of time with no preposition at all.

So instead of saying something like for half an hour, Russian simply says:

• полчаса — half an hour
• час — an hour
• два часа — two hours

Examples:

• Я работаю час. — I’ve been working for an hour.
• Мы ждём уже два часа. — We’ve already been waiting for two hours.

So in Я жду такси уже полчаса, the phrase уже полчаса means for half an hour already.

Yes. Russian word order is fairly flexible.

The original sentence:

• Я жду такси уже полчаса.

is completely natural and neutral.

But you could also say:

• Я уже полчаса жду такси.
• Такси я жду уже полчаса.

These all mean roughly the same thing, but the emphasis changes:

• Я уже полчаса жду такси emphasizes the length of time
• Такси я жду уже полчаса emphasizes the taxi

So the original version is good, but it is not the only possible order.

Yes, you could.

Ожидать also means to wait for / to expect, so:

• Я ожидаю такси уже полчаса.

is grammatically fine.

But ждать is more common and natural in everyday speech. Ожидать sounds a bit more formal or bookish in many contexts.

So for normal conversation, Я жду такси уже полчаса is the best choice.