Kedinin patisi incinmiş olsa bile yine de mırlıyordu.

This is the normal Turkish possessive pattern for X's Y.

• kedi = cat
• kedinin = of the cat / the cat's
• pati = paw
• patisi = its paw / his paw / her paw

So:

• kedinin patisi = the cat's paw

In Turkish, the possessor usually takes the genitive ending, and the thing possessed takes a possessive ending:

• kedi-nin pati-si

This is very common:

• adamın arabası = the man's car
• evin kapısı = the house's door / the door of the house

Because the noun is possessed.

Here, pati is not just a paw in a general sense. It means the cat's paw, so Turkish marks that with the 3rd person possessive suffix.

• pati = paw
• patisi = its paw

The -s- is a buffer consonant that appears because pati ends in a vowel.

So:

• kedi pati = not correct for the cat's paw
• kedinin patisi = correct

İncinmiş comes from incinmek, which means to get hurt, to be injured, or sometimes to be emotionally hurt, depending on context.

Here it is physical:

• incinmek = to get injured / to be hurt
• incinmiş = hurt, injured, apparently hurt, has gotten hurt

The ending -miş often adds a sense like:

• the speaker learned it indirectly,
• or is inferring it,
• or is focusing on the resulting state.

In this sentence, it helps create the idea being in an injured state.

So kedinin patisi incinmiş means something like:

• the cat's paw was injured
• the cat's paw had gotten hurt

Olsa bile is a very common concessive structure meaning:

• even if ...
• even though ...
• although ...

It comes from:

• olsa = if it were / even if it is, from olmak
• bile = even

So:

• incinmiş olsa bile = even if it was injured / even though it was injured

This structure often follows a noun, adjective, or participle phrase:

• zor olsa bile = even if it is difficult
• geç kalmış olsa bile = even if he/she was late
• hasta olsa bile = even if he/she is sick

In your sentence, it creates a contrast with the main clause.

Both can often be translated as even if it gets/got hurt or even though it was hurt, but there is a nuance.

• incinse bile = even if it gets hurt / even if it were hurt
• incinmiş olsa bile = even if it was already hurt / even though it had become injured

The version with -miş olsa bile sounds a little more like the injury is an already existing condition or completed state.

So:

• incinse bile focuses more on the possibility/condition
• incinmiş olsa bile focuses more on the state of having been injured

In many contexts, both are possible, but incinmiş olsa bile sounds a bit more literary or more carefully framed.

Yine de means:

• still
• nevertheless
• all the same

You are right that bile already gives an even if / although meaning. But yine de strengthens the contrast in the main clause.

So the sentence works like this:

• incinmiş olsa bile = even though its paw was injured
• yine de mırlıyordu = it was still purring

English often does the same:

• Even though it was hurt, it still purred.

So yine de is not strictly necessary, but it makes the contrast clearer and more natural.

Because mırlıyordu is the past continuous / imperfective form:

• mırlamak = to purr
• mırlıyor = is purring
• mırlıyordu = was purring

This suggests an ongoing action in the past.

So:

• mırlıyordu = it was purring
• mırladı = it purred

The sentence is not just saying the cat made one completed purring action. It presents the purring as something continuing despite the injury.

That is why mırlıyordu fits very well.

The understood subject is the cat.

Turkish often leaves out pronouns and even repeated subjects when they are clear from context. Since the sentence begins with kedinin patisi and the whole scene is about the cat, the listener naturally understands that the one purring is the cat.

So English needs it, but Turkish does not:

• Kedinin patisi incinmiş olsa bile yine de mırlıyordu.
• Even though the cat's paw was injured, it was still purring.

If you wanted, you could make the subject more explicit:

• Kedinin patisi incinmiş olsa bile kedi yine de mırlıyordu.

But that is usually unnecessary.

Yes, but it is very normal Turkish word order.

A common pattern is:

subordinate clause + main clause

So here:

• Kedinin patisi incinmiş olsa bile = even though the cat's paw was injured
• yine de mırlıyordu = it was still purring

Also, Turkish usually puts the verb near the end of its clause, so:

• mırlıyordu comes at the end

You can move some elements for emphasis, especially yine de, but this version is very natural and standard.

Not necessarily in a strong literal sense.

It is true that -miş is often called the reported or inferential past. It can mean:

• the speaker heard it from someone else,
• or inferred it from evidence,
• or is describing a resulting state.

But in real Turkish, -miş is often used more flexibly than a simple grammar label suggests.

In incinmiş olsa bile, the form is doing more than just saying I didn’t see it myself. It helps build the concessive phrase and presents the paw as being in an injured state.

So a good practical understanding here is:

• incinmiş = injured / having been hurt

You do not need to force a strong apparently meaning every time you see -miş.

Yes, depending on context, there can be a slight shade of uncertainty or tentativeness.

Because of -miş and olsa bile, the phrase can sound a bit like:

• even if it was injured
• even though it was injured
• even if it seems to have been injured

But in many contexts, the most natural English translation will simply be:

• Even though the cat's paw was injured, it was still purring.

So the exact English wording depends on how much uncertainty the surrounding context suggests.

Because kedinin means the cat's, while kedisinin would usually mean his/her/its cat's.

Compare:

• kedi = cat
• kedinin = the cat's
• kedisi = his/her/its cat
• kedisinin = his/her/its cat's

So:

• kedinin patisi = the cat's paw
• kedisinin patisi = his/her cat's paw

In your sentence, we are talking directly about the cat, so kedinin is the correct form.