Регистрационный взнос уже оплачен.

Because оплачен is a short passive participle, not a regular past-tense verb.

In this sentence, Russian is focusing on the state/result: the fee is / has been paid.

• Регистрационный взнос уже оплачен. = The registration fee has already been paid.
• Регистрационный взнос уже оплатили. = They already paid the registration fee.

So:

• оплатили tells you that someone performed the action
• оплачен tells you the result of that action

Russian often uses this structure where English would use has been + past participle.

Russian normally does not use a present-tense form of “to be” in sentences like this.

So:

• Регистрационный взнос уже оплачен. literally looks like:
• Registration fee already paid

But in natural English, you translate it as:

• The registration fee is already paid or more naturally
• The registration fee has already been paid

The missing is is completely normal in Russian.

It is in the nominative case.

That is because регистрационный взнос is the grammatical subject of the sentence — the thing being described.

Breakdown:

• регистрационный = nominative masculine singular adjective
• взнос = nominative masculine singular noun

The predicate оплачен agrees with взнос in gender and number:

• masculine singular noun → оплачен

That ending shows it is a short-form passive participle, masculine singular.

Here are the matching forms:

• masculine: оплачен
• feminine: оплачена
• neuter: оплачено
• plural: оплачены

Examples:

• Счёт оплачен. = The bill has been paid.
• Пошлина оплачена. = The duty/fee has been paid.
• Место оплачено. = The place/seat has been paid for.
• Счета оплачены. = The bills have been paid.

So in your sentence, since взнос is masculine singular, the correct form is оплачен.

Уже means already.

So:

• Регистрационный взнос оплачен. = The registration fee has been paid.
• Регистрационный взнос уже оплачен. = The registration fee has already been paid.

It marks that the action/result happened before now, often earlier than expected or before some other step.

That is the most neutral word order here.

• Регистрационный взнос уже оплачен.

Russian word order is flexible, but placement changes emphasis.

For example:

• Регистрационный взнос уже оплачен.
  Neutral: the fee is already paid.

• Уже оплачен регистрационный взнос.
  More marked/literary; emphasizes already paid.

• Регистрационный взнос оплачен уже.
  Possible in speech, often with a certain intonation; can sound contrastive or emphatic.

For a learner, the original word order is the safest and most natural.

Регистрационный is an adjective meaning registration-related or for registration.

It comes from the noun:

• регистрация = registration

So:

• регистрационный взнос = registration fee
• literally: registration-related contribution/payment

This adjective is common in administrative or formal language, for example:

• регистрационная форма = registration form
• регистрационный номер = registration number

Взнос literally means a payment, contribution, installment, or fee, depending on context.

In administrative or formal contexts, взнос often refers to a required payment, such as:

• членский взнос = membership fee
• вступительный взнос = entrance fee / initiation fee
• регистрационный взнос = registration fee

It is a more specific word than just деньги or плата. It suggests a formal payment made as part of membership, participation, registration, and so on.

Russian commonly uses оплатить / оплачивать with a direct object for things like bills, fees, and services:

• оплатить счёт = to pay the bill
• оплатить взнос = to pay the fee/contribution

That is why the passive/result form is:

• взнос оплачен

By contrast, заплатить often appears with:

• an amount: заплатить 100 рублей
• or за + accusative: заплатить за регистрацию

So these are both natural, but they are built differently:

• Регистрационный взнос уже оплачен.
• За регистрацию уже заплатили.

The first one is the better match for your sentence.

Yes, it sounds fairly formal or administrative.

You might see it in:

• an email from an organizer
• a website account page
• a payment confirmation
• official event paperwork

In everyday conversation, people might say something a bit less formal, depending on context, but this sentence is perfectly natural in any official setting.

Yes.

Russian does not always force the same tense distinction that English does here. In context, оплачен can correspond to:

• is already paid
• has already been paid

In most real situations, English would naturally use:

• The registration fee has already been paid.

That is usually the best translation.

Yes, but it means something slightly different.

• Регистрационный взнос уже оплачен.
  The registration fee is already paid / has already been paid.

• Регистрационный взнос уже был оплачен.
  The registration fee had already been paid / was already paid.

Adding был makes the sentence more clearly about a past reference point. It is useful if you are describing what was true at some moment in the past.

So if you are just stating the current status, the original sentence without был is better.

A helpful stress marking is:

Регистрацио́нный взно́с уже́ опла́чен.

Approximate pronunciation:

• reh-gyee-strah-tsee-ON-nyy vznos oo-ZHEH ah-PLAH-chen

A few notes:

• взнос is one syllable cluster at the start: vz-
• уже́ has stress on the second syllable
• опла́чен has stress on пла

If pronunciation feels difficult, practice it in chunks:

• регистрационный взнос
• уже оплачен

Then combine them.

Yes. The short participle must agree in gender and number with the noun it describes.

Compare:

• Взнос оплачен. — masculine
• Пошлина оплачена. — feminine
• Место оплачено. — neuter
• Взносы оплачены. — plural

So agreement works much like adjective agreement, except this is a short-form participle used as the predicate.