Эту операцию можно оспорить.

Because эту is the accusative singular feminine form of эта.

In Эту операцию можно оспорить, the noun операцию is the direct object of оспорить, so the demonstrative this has to agree with it in:

• gender: feminine
• number: singular
• case: accusative

So:

• nominative: эта операция = this operation
• accusative: эту операцию = this operation as an object

For the same reason: it is in the accusative case.

The dictionary form is операция.
Feminine nouns ending in -ия usually change like this in the singular:

• nominative: операция
• accusative: операцию

Here it is the thing being challenged, so Russian marks it as the direct object.

Можно means it is possible, it is allowed, or more naturally here, can be.

So можно + infinitive is a very common Russian pattern meaning:

• it is possible to ...
• one can ...
• ... can be ... in natural English

So:

• Эту операцию можно оспорить
  = This operation can be challenged = literally, It is possible to challenge this operation

Russian uses this impersonal structure very often where English might use can.

Not formally, no.

In English, This operation can be challenged sounds passive.
But in Russian, Эту операцию можно оспорить is built differently:

• можно = impersonal it is possible
• оспорить = active infinitive to challenge
• эту операцию = direct object

So the literal structure is closer to:

• This operation, it is possible to challenge

Russian often uses this kind of impersonal construction instead of a passive one.

Оспорить is perfective, while оспаривать is imperfective.

Here, the perfective verb is natural because the sentence talks about the possibility of completing a single action:

• оспорить = to challenge successfully / to contest as a whole
• оспаривать = to be challenging / to challenge repeatedly or as a process

With можно + infinitive, Russian often uses the perfective when it means it is possible to do this action in a specific case.

So:

• Эту операцию можно оспорить = This operation can be challenged
• Эту операцию можно оспаривать would sound less natural here and would suggest an ongoing or repeated process.

The dictionary form here is already оспорить.

Its aspect pair is:

• оспаривать = imperfective
• оспорить = perfective

This is a common thing to learn with Russian verbs: many verbs come in aspect pairs, and you usually need to know both.

Yes, you absolutely could say Можно оспорить эту операцию.

Both are grammatical, but the word order changes the focus a bit.

• Эту операцию можно оспорить
  puts this operation up front, making it the topic or focus
• Можно оспорить эту операцию
  sounds a bit more neutral: It is possible to challenge this operation

Russian word order is more flexible than English word order, and it is often used to show emphasis or information structure rather than basic grammar.

Exactly. The sentence is impersonal, so it does not name the person who can do the action.

It simply says that challenging the operation is possible.

Depending on context, it could mean:

• someone can challenge it
• one can challenge it
• it can be challenged
• the customer / bank / lawyer can challenge it if that is understood from context

If you want to name the person, Russian would usually say so directly, for example:

• Клиент может оспорить эту операцию. = The client can challenge this transaction.
• Мы можем оспорить эту операцию. = We can challenge this transaction.

Yes, можно can mean either it is possible or it is allowed, depending on context.

For example:

• Здесь можно курить. = You may smoke here / It is allowed to smoke here
• Эту дверь можно открыть. = This door can be opened / It is possible to open this door

In Эту операцию можно оспорить, the most natural meaning is it is possible to challenge it, not really you are permitted to challenge it, although legal or procedural permission could be part of the context.

The stress is:

• э́ту
• опера́цию
• мо́жно
• оспо́рить

So you can mark it as:

Э́ту опера́цию мо́жно оспо́рить.

A rough pronunciation guide:

• эту ≈ EH-too
• операцию ≈ a-pye-RA-tsi-yu
• можно ≈ MOZH-na
• оспорить ≈ as-PO-rʲitʹ

No. In English that would mean the operation itself is doing the challenging, which is not what the Russian says.

Even though Russian uses an active infinitive, the meaning is:

• This operation can be challenged
• It is possible to challenge this operation

So the natural English translation is passive, even though the Russian grammar is not a standard passive construction.