huninkyoseisyuzyutu to wakutinsessyu ga owatta ka kakuninsuru.

It is a compound noun meaning spay/neuter surgery or sterilization surgery, especially for animals.

Breakdown:

• 不妊 = infertility / sterilization
• 去勢 = castration / neutering
• 手術 = surgery

As one chunk, it refers to the surgical procedure done to prevent reproduction. In context, it usually means the appropriate sterilization surgery for the animal.

と connects nouns and means and.

So:

• 不妊去勢手術 と ワクチン接種
• spay/neuter surgery and vaccination

Here it is listing both procedures clearly and completely. This is different from や, which can imply a non-exhaustive list like A and B and so on.

Because 不妊去勢手術とワクチン接種 is the subject of 終わった.

In other words, the embedded clause is:

• 不妊去勢手術とワクチン接種が終わった
• the surgery and vaccination were completed

So even though the whole sentence means check whether ..., inside the smaller clause, Japanese treats the procedures as the thing that finished / were completed.

終わった is the plain past form of 終わる: to finish / to be finished.

Here it means was completed or has finished. Japanese often uses the past form when talking about whether something has already been completed.

So:

• 終わったか = whether it is finished / whether it has been completed

This is very natural in a checklist or confirmation context.

Yes, it could.

• 終わったか確認する = check whether it finished / was completed
• 終わっているか確認する = check whether it is in the finished state

In many real situations, both are natural.
終わっている puts a little more focus on the current state, while 終わった focuses on the completion event. In everyday use, the difference may be small.

Here, か marks an embedded question, so it means whether or if.

• 終わったか確認する
• check whether it has been completed

This is not the か used at the end of a direct question like 行きますか.
Instead, it turns the preceding clause into the content being checked.

Because a clause ending in か can directly act as the thing being confirmed.

So this is normal:

• 終わったか確認する

You may also see:

• 終わったかを確認する

That version is also possible and a little more explicit, but the を is often omitted in natural Japanese.

確認する is the plain form. It is common in:

• written instructions
• manuals
• checklists
• note-style sentences

So this sentence sounds like a neutral instruction such as Confirm whether the surgery and vaccination have been completed.

If you wanted a polite spoken version, you could say:

• 不妊去勢手術とワクチン接種が終わったか確認します。

Yes, it is relatively formal or technical.

• ワクチン接種 = vaccination / vaccine administration

A more casual way to say it would be something like:

• ワクチンを打つ
• to get/give a vaccine shot

In medical, veterinary, or administrative language, ワクチン接種 is very common.

Yes. Because the two nouns are joined by と, the sentence is asking about both items together.

So the idea is:

• check whether the spay/neuter surgery and the vaccination have been completed

If one has been done but the other has not, then the answer to the overall check would still be not yet.

That is normal in Japanese. The subject is omitted because it is understood from context.

Depending on the situation, it could mean:

• a veterinarian checks
• clinic staff check
• the pet owner checks
• someone reviewing a record checks

Japanese often leaves out the subject when it is obvious or unimportant.

Yes.

• 不妊去勢手術とワクチン接種が終わったかどうか確認する

This means almost the same thing: check whether or not the surgery and vaccination have been completed.

Using just か is already very natural here, so かどうか is not necessary, but it can sound a little more explicit.