Ветеринар оглянув лапу.

Because лапу is in the accusative singular, which is the case used for a direct object.

• Dictionary form: лапа = paw
• In this sentence, the paw is what the vet examined, so it becomes the direct object.

• For many feminine nouns ending in -а, the accusative singular ends in -у:

• лапа → лапу
• книга → книгу
• мама → маму

So Ветеринар оглянув лапу literally has:

• Ветеринар = subject
• оглянув = examined
• лапу = direct object

Ukrainian shows this mainly through case endings, not just word order.

• ветеринар is in the nominative case, which is the normal case for the subject.
• лапу is in the accusative case, marking it as the direct object.

So even if the order changed, the endings would still help you understand who did what.

Оглянув is the past tense, masculine singular form of the verb.

In Ukrainian past tense, the verb agrees with the subject in gender and number.

Since ветеринар is one male person, the verb is masculine singular:

• оглянув = he examined

Compare:

• ветеринар оглянув = the male vet examined
• ветеринарка оглянула = the female vet examined
• ветеринари оглянули = the vets examined

Оглянув is perfective.

That means the action is seen as completed: the vet examined the paw and finished doing so.

The verb pair is:

• оглядати = imperfective, to be examining / to examine in a general or repeated sense
• оглянути = perfective, to examine and complete the examination

So:

• Ветеринар оглядав лапу suggests an ongoing process: The vet was examining the paw.
• Ветеринар оглянув лапу suggests a completed act: The vet examined the paw.

Because Ukrainian does not have articles like the or a/an.

So ветеринар can mean:

• the veterinarian
• a veterinarian

And лапу can mean:

• the paw
• a paw

Which one sounds best depends on context. English has to choose; Ukrainian usually does not.

Yes. Ukrainian word order is fairly flexible because cases do a lot of the grammatical work.

The neutral order here is:

• Ветеринар оглянув лапу.

But you could also hear:

• Лапу оглянув ветеринар.
• Оглянув ветеринар лапу.

These versions may sound more marked or emphasize different parts of the sentence, but the basic meaning stays the same because of the case forms.

Usually it means something stronger than just looked at. It often means:

• examined
• inspected
• looked over

So in a medical or veterinary context, оглянув is a very natural verb for checking a patient or body part.

Because лапу is a direct object, and direct objects normally do not need a preposition in Ukrainian.

English also does this with many verbs:

• The vet examined the paw.

Not:

• The vet examined at the paw.

Same idea in Ukrainian:

• оглянув лапу = examined the paw

The dictionary form is оглянути.

In dictionaries, Ukrainian verbs are usually listed in the infinitive form, which often ends in -ти.

So:

• оглянути = to examine / to inspect
• sentence form: оглянув = examined

Then the past tense verb would change to agree with a feminine subject.

You would say:

• Ветеринарка оглянула лапу.

or, depending on wording,

• Жінка-ветеринар оглянула лапу.

The important change is:

• masculine: оглянув
• feminine: оглянула

A useful stress guide is:

• ветерина́р
• огляну́в
• лапу́

So the whole sentence is approximately:

• veteryNAHR ohlyaNOOV laPOO

That is only an approximation for English speakers, but the main thing to notice is that Ukrainian stress is important and is not always predictable.

Yes. Its main meaning is paw, especially for an animal.

Depending on context, it can sometimes be understood more loosely as:

• foot
• leg
• even colloquially something like hand in certain informal uses

But in a sentence about a veterinarian, paw is the most natural interpretation.

Normally, no.

For a human leg, you would usually use нога:

• Лікар оглянув ногу. = The doctor examined the leg.

Лапа is mainly for animals, so in this sentence it strongly suggests an animal patient.