Эта традиция существует уже век.

Because традиция is a feminine singular noun, so the word for this has to agree with it.

• masculine: этот
• feminine: эта
• neuter: это
• plural: эти

So:

• эта традиция = this tradition

Here эта традиция is in the nominative singular because it is the subject of the sentence.

Russian often uses the present tense where English uses the present perfect.

So:

• Эта традиция существует уже век. literally = This tradition exists already a century natural English = This tradition has existed for a century

The idea is: the tradition began in the past and still exists now. Russian commonly expresses that with:

• a present-tense verb
• plus a word like уже
• plus a duration expression such as век, год, много лет

So this is completely normal Russian.

Существовать is a full verb meaning to exist. It works well when talking about something’s continued existence over time.

• существует = exists
• есть in modern Russian is often used more like there is/there are or is omitted entirely in simple X is Y sentences

So:

• Эта традиция существует уже век = natural
• Эта традиция есть уже век = not natural here

Using существует emphasizes that the tradition is a real thing that has continued to exist.

Yes, уже basically means already, but in this kind of sentence it often has the sense of:

• by now
• for ... now
• as long as ... already

So уже век does not just mean already a century word-for-word; it means something like:

• for a century now
• has been around for a century already

It highlights the long duration up to the present moment.

Russian often expresses duration of time with a noun in the accusative case and no preposition.

So English:

• for a century

becomes Russian:

• век

This is a common pattern:

• ждать час = to wait for an hour
• работать весь день = to work all day
• жить год = to live for a year

In your sentence, век is functioning as a duration expression.

It is accusative singular, used to show length of time.

The tricky part is that for many inanimate masculine nouns, the accusative singular looks exactly like the nominative singular.

So:

• nominative: век
• accusative: век

They look the same, but the function is different. Here it is not the subject; it tells you how long the tradition has existed.

It can be understood in two ways depending on context and style:

1. literally: a century
2. more loosely/stylistically: for ages, for a very long time

In this sentence, most learners should understand it first as:

• for a century

But it also has a slightly literary or elevated feel compared with a more neutral phrase like сто лет.

Yes, absolutely. That is very natural Russian.

• Эта традиция существует уже век.
• Эта традиция существует уже сто лет.

The difference is mainly stylistic:

• уже сто лет = more neutral, straightforward, explicit
• уже век = shorter, slightly more literary or compact

If you want to be clearly factual and exact, сто лет is often the safer choice.

Yes. Russian word order is flexible, and changing it usually changes emphasis, not the core meaning.

Possible versions:

• Эта традиция существует уже век.
  neutral, natural
• Уже век существует эта традиция.
  stronger emphasis on the long duration
• Эта традиция уже век существует.
  also possible, with emphasis on уже век

The version you were given is a very standard, clear order: subject + verb + time expression.

Yes. Существовать is imperfective, and that fits the meaning perfectly.

Imperfective verbs are used for:

• ongoing states
• repeated actions
• actions viewed as continuing or uncompleted

Here the tradition is not something that finished; it is something that has been existing and still exists. So the imperfective is exactly what Russian needs.

That is another reason the sentence uses the present tense so naturally: it describes a continuing state.