Нам пришлось допечь пирог, потому что середина была ещё сырой.

Because прийтись / пришлось in the sense of to have to is usually used in an impersonal construction in Russian.

• нам = to us / for us (dative case)
• пришлось = it turned out that it was necessary / we had to

So нам пришлось допечь пирог literally feels like:

• to us, it became necessary to finish baking the pie

Russian does not say мы пришлось here, because мы is nominative, and this pattern requires the person affected to be in the dative.

Very common pattern:

• мне пришлось уйти = I had to leave
• тебе пришлось ждать = you had to wait
• нам пришлось допечь пирог = we had to finish baking the pie

Because this use of прийтись is impersonal. In impersonal expressions, Russian often uses a default past tense form in neuter singular:

• пришлось
• удалось
• повезло
• не удалось

There is no normal subject in the nominative here, so Russian uses the neuter singular form by default.

So even though нам means we/us, the verb does not agree with нам.

Here пришлось means had to, was forced to, or ended up having to.

It often suggests that the action was necessary because of circumstances, not because the speaker wanted to do it.

So:

• Нам пришлось допечь пирог
  = We had to finish baking the pie

This can sound a little more circumstantial than a simple English had to:

• the pie was not ready
• because of that, it became necessary to bake it longer

Допечь means to finish baking or to bake a bit more until done.

It comes from:

• печь = to bake
• prefix до- = to do something to the end / complete it / continue until the proper result

So допечь пирог means:

• the pie was already in the oven
• it was partly baked
• but not fully done
• so it needed more baking

That is why допечь is better than just печь here.
Using печь would sound more like to bake a pie in general, not to finish baking one that is still underdone.

It is perfective because the sentence is about a completed action: they baked it a bit more and brought it to the proper result.

Perfective verbs in Russian often focus on:

• a single completed action
• reaching a result

Here the result is important:

• the pie needed more baking
• they finished that extra baking

The imperfective partner would be допекать. That would be used in contexts like:

• repeated action
• process
• background description

For example:

• Нам пришлось допекать пирог ещё десять минут.
  = We had to keep baking the pie for another ten minutes.

But in your sentence, допечь fits because the action is viewed as a whole.

Пирог is the direct object of допечь, so it is in the accusative case.

For an inanimate masculine noun like пирог, the accusative singular looks exactly the same as the nominative singular:

• nominative: пирог
• accusative: пирог

So the form does change by case grammatically, but it just happens to look the same.

Compare with a feminine noun:

• допечь булку
  Here you can see the accusative form more clearly.

Because середина is a feminine noun.

In the past tense, Russian verbs agree with the grammatical gender of the subject:

• masculine: был
• feminine: была
• neuter: было
• plural: были

And adjectives also agree with the noun:

• сырая середина = a raw/underdone middle
• середина была сырой = the middle was raw/underdone

So:

• середина = feminine
• therefore была and сырой are feminine too

Yes, сырой literally means raw or uncooked, but in cooking it can also mean underdone or not baked through.

In this sentence:

• середина была ещё сырой means
• the middle was still raw / still underdone / not fully baked

For a pie or cake, English often says:

• the middle was still raw
• the center was still undercooked
• the middle wasn’t done yet

So сырой is very natural here.

Here ещё means still.

• середина была ещё сырой
  = the middle was still raw

This use of ещё is common when something has not changed yet, but you expect it should.

Compare:

• Он ещё спит. = He is still sleeping.
• Я ещё не готов. = I am not ready yet / still not ready.
• Пирог ещё сырой. = The pie is still underdone.

So here it emphasizes that, even at that point, the center had not finished baking.

Because after была, the adjective is in the instrumental case, not nominative.

This is a very common pattern in Russian:

• X был/была/было Y-ом / Y-ой / Y-им

So:

• nominative: сырая
• instrumental feminine singular: сырой

That is why:

• середина была сырая is possible in some contexts, especially in more colloquial speech
• but середина была сырой is the standard textbook pattern after была

So the sentence uses a very normal structure:

• середина была сырой

Yes, very strongly.

That is one of the main ideas of the prefix до- here. It suggests:

• the action had already started
• but had not reached the proper end
• so it needed to be completed

So допечь пирог is not the same as simply bake a pie from the beginning. It means more like:

• finish baking the pie
• bake the pie a bit longer until it’s done

Yes. Russian word order is flexible, and changing it usually changes emphasis, not the basic meaning.

Original:

• Нам пришлось допечь пирог, потому что середина была ещё сырой.

Possible variations:

• Пирог нам пришлось допечь, потому что середина была ещё сырой.
  Emphasis on the pie
• Потому что середина была ещё сырой, нам пришлось допечь пирог.
  Emphasis on the reason first

The original order is very natural and neutral:

1. what happened
2. why it happened

Yes. Потому что most directly means because, but in English the best translation may sometimes be since, depending on style.

• because the middle was still raw
• since the middle was still raw

Both work, though because is the most direct equivalent.

Russian потому что is one of the most common ways to introduce a reason.

Yes. Середина is more natural here.

• середина = the middle, the inside middle part
• центр = the center, often more geometric or technical

When talking about food, especially whether something is baked through, Russian commonly uses середина:

• середина пирога
• середина торта
• середина кекса

So середина была ещё сырой sounds very natural for cooking.