Раз уж ты всё равно хочешь новую полку, лучше сразу проверить её высоту и глубину.

Раз уж is a conversational way to mean something like:

• since
• given that
• if you’re going to ... anyway
• as long as ...

In this sentence, Раз уж ты всё равно хочешь новую полку means something like:

Since you want a new shelf anyway...

It often introduces a fact that the speaker treats as already decided, and then gives a logical consequence or recommendation.

Compare:

• Раз уж пришёл, оставайся. — Since you’ve come, stay.
• Раз уж начал, закончи. — Since you’ve started, finish.

The particle уж adds a slightly emphatic, natural spoken tone.

Yes, всё равно can literally mean it makes no difference / it’s all the same, but very often in real Russian it also means:

• anyway
• in any case
• regardless

So here:

ты всё равно хочешь новую полку

means:

you want a new shelf anyway

The idea is: whether or not something else is true, the person already wants the shelf, so it makes sense to do the next thing.

Examples:

• Я всё равно пойду. — I’m going anyway.
• Он всё равно это купит. — He’ll buy it anyway.

Because хотеть takes a direct object, and here полка is that object.

The basic dictionary form is:

• новая полка — a new shelf

But after хочешь (you want), it has to go into the accusative case:

• новую полку

This is because both the adjective and the noun agree in case, gender, and number.

So:

• новая полка — nominative
• новую полку — accusative singular feminine

Same pattern:

• Я вижу новую машину. — I see a new car.
• Он хочет хорошую работу. — He wants a good job.

Because проверить takes direct objects too, and the things being checked are:

• высота — height
• глубина — depth

As direct objects, they go into the accusative singular:

• высота → высоту
• глубина → глубину

So:

• проверить её высоту и глубину
  = to check its height and depth

The word её here means its (or literally her, depending on context). Since полка is feminine in Russian, the pronoun referring back to it is её.

Because свой normally refers back to the subject of the clause.

Here, the implied subject of проверить is the person who should do the checking, while её refers to полку (the shelf), not to that subject.

So:

• её высоту = its height (the shelf’s height)

If you used свою, it would suggest one’s own height, which makes no sense here.

A helpful rule:

• use свой when something belongs to the subject
• use его / её / их when it belongs to someone/something else

This is a very common Russian pattern. Лучше can be used impersonally to mean:

• it’s better to...
• you’d better...

So:

лучше сразу проверить means it’s better to check right away / you’d better check right away

Russian often leaves out words that English needs. There is no explicit it is here, but the meaning is understood.

Compare:

• Лучше подождать. — It’s better to wait.
• Лучше не спорить. — Better not argue.

Because the sentence is talking about one complete checking action.

• проверить = perfective = to check, to verify, to make sure as a completed act
• проверять = imperfective = to be checking / to check regularly / to check in general

Here the meaning is: before buying or choosing the shelf, do one check of the measurements. That is a completed action, so проверить fits naturally.

Compare:

• Нужно проверить размеры. — You need to check the dimensions.
• Он всегда проверяет размеры. — He always checks the dimensions.

Сразу means:

• right away
• at once
• immediately
• sometimes from the start

In this sentence, it suggests: don’t wait until later; check the measurements as soon as you’re dealing with the new shelf.

So:

лучше сразу проверить = it’s better to check right away

Russian word order is flexible, and this version sounds natural and neutral.

• лучше sets up the recommendation
• сразу modifies проверить
• проверить comes at the end of that phrase as the main action

So the structure flows as:

Since..., it’s better right away to check...

Other word orders are possible, but they may sound more marked or place emphasis differently. For example:

• лучше проверить сразу — also possible, with a slightly different rhythm
• сразу лучше проверить — possible in some contexts, but more marked

For learners, the given order is a very standard one.

Its basic meaning is shelf.

Depending on context, it could be:

• a wall shelf
• a shelf in furniture
• sometimes a rack or storage shelf

Here, because the sentence talks about height and depth, it clearly means a physical shelf or shelving unit whose dimensions matter.

Because the first part is a subordinate clause introduced by раз уж:

• Раз уж ты всё равно хочешь новую полку, ...
• Since you want a new shelf anyway, ...

Russian separates that introductory clause from the main clause with a comma.

So the sentence structure is:

1. subordinate/background clause
2. main recommendation

This is very similar to English punctuation in Since you want a new shelf anyway, ...

Yes, in context it often does.

Проверить literally means to check / verify, but when talking about dimensions like height and depth, it can naturally imply:

• checking the measurements you already have
• confirming them
• even measuring them to make sure

So depending on context, an English translation might be:

• check its height and depth
• make sure of its height and depth
• measure its height and depth

The Russian itself stays a bit broader than a specific verb like measure.