Рана на лапі під кігтем.

It is better understood as a short note, caption, or medical/veterinary-style record rather than a full everyday sentence.

Ukrainian often allows very compact wording, especially in descriptions. Here the idea is simply being labeled:

• Рана = the main item being identified
• на лапі під кігтем = where it is

A fuller version could be:

• На лапі під кігтем є рана.
• У тварини рана на лапі під кігтем.

So the original sounds natural as a brief descriptive note.

Because на here means on / on the paw in the sense of location, and with that meaning it normally takes the locative case.

The dictionary form is:

• лапа = paw

But after на for location, it becomes:

• на лапі = on the paw

Compare:

• лапа = paw
• на лапі = on the paw
• на лапу = onto the paw / to the paw

So лапі is there because the sentence is describing where the wound is.

Because after під meaning under / beneath in a static location, Ukrainian uses the instrumental case.

The dictionary form is:

• кіготь = claw

But here it becomes:

• під кігтем = under the claw

So:

• кіготь = nominative
• кігтем = instrumental singular

There is also a stem change here that learners often notice:

• кіготь
• кігтя
• кігтем

That disappearing о is normal for this noun in oblique forms.

Because під changes case depending on whether you mean location or movement.

In this sentence, the wound is already located under the claw, so Ukrainian uses:

• під + instrumental = under, beneath, while being there

Example pattern:

• під столом = under the table

If something moves to a position under something, Ukrainian can use:

• під + accusative = to under / underneath

Example:

• залізти під стіл = to crawl under the table

So in під кігтем, the instrumental marks a fixed location.

Because рана is the main noun being named.

This kind of short descriptive phrase often starts with the thing being identified, and that noun stays in its basic form, the nominative singular:

• рана = wound

If you expand the idea into a fuller sentence, it still makes sense as the main noun:

• На лапі під кігтем є рана.

So рана is nominative because it is the central item being pointed out.

Usually, no. Лапа normally means an animal paw.

For humans, Ukrainian normally uses words like:

• рука = hand/arm
• нога = leg/foot
• ступня = foot

If лапа is used about a person, it often sounds:

• humorous
• childish
• affectionate
• rude or mocking

So in this sentence, лапа strongly suggests that the wound is on an animal.

This is a very common question.

• кіготь = claw, talon
• ніготь = fingernail / toenail

With лапа, кіготь is the natural choice, because animals have claws on their paws.

So:

• лапа + кіготь sounds natural for an animal
• ніготь would usually be used for a person’s nail

Yes. Ukrainian word order is fairly flexible.

The given order:

• Рана на лапі під кігтем.

puts the focus first on the wound.

Other possible orders include:

• На лапі під кігтем рана.
  More sentence-like, with the location first.

• Під кігтем на лапі рана.
  Stronger emphasis on the exact location.

The original version sounds natural if you are labeling or noting the injury itself.

You do not know from grammar alone, because Ukrainian has no articles like a and the.

So рана can mean:

• a wound
• the wound
• just wound as a label in a note

The exact English choice depends on context, not on a special Ukrainian word.