W lodówce jest jeszcze budyń czekoladowy, więc nie muszę dziś piec ciasta.

Because after w meaning in for a fixed location, Polish uses the locative case.

The dictionary form is:

• lodówka = fridge

But after w for location, it becomes:

• w lodówce = in the fridge

So this is a case change, not a different word.

No. W can take different cases depending on meaning.

Here it means location, so it takes the locative:

• w lodówce = in the fridge

With movement into something, Polish often uses a different construction, very commonly do + genitive:

• do lodówki = into/to the fridge

So the case depends on whether you are talking about where something is or where something is going.

Because Polish does not need a dummy subject like English there.

A sentence like:

• W lodówce jest budyń

literally looks more like:

• In the fridge is pudding

but it functions as normal Polish for There is pudding in the fridge.

So jest often covers the idea of there is, especially when a location is given.

Here jeszcze means still.

It suggests that the chocolate pudding is still there / remains there / hasn’t been used up yet.

That is a very common use of jeszcze:

• jeszcze jest = there still is

In other contexts, jeszcze can also mean things like more, another, or yet, so context matters a lot.

Yes, Polish word order is fairly flexible.

This order is natural because it starts with the place:

• W lodówce = in the fridge

That sets the scene first, and then the sentence tells you what is there.

So the structure feels like:

1. location
2. existence
3. item

That is a very normal way to present information in Polish. Other word orders are possible, but this one sounds smooth and natural.

Czekoladowy is an adjective agreeing with budyń.

• budyń is masculine singular
• so the adjective must also be masculine singular
• that gives czekoladowy

So the ending -y is there because the adjective has to match the noun grammatically.

Also, the order noun + adjective is very common in food/product names in Polish:

• budyń czekoladowy
• jogurt truskawkowy
• kisiel malinowy

You may also see adjective-first order in Polish, but this noun-first pattern is very natural for kinds of products.

Because Polish usually leaves out subject pronouns when the verb ending already makes the subject clear.

• muszę already means I must / I have to

So ja is unnecessary unless you want emphasis or contrast.

Compare:

• Nie muszę dziś piec ciasta. = neutral
• Ja nie muszę dziś piec ciasta. = I don’t have to, maybe unlike someone else

This is one of the biggest differences from English.

After verbs like musieć (must / have to), Polish normally uses the infinitive of the second verb.

So:

• muszę piec = I have to bake
• muszę iść = I have to go
• muszę pracować = I have to work

This is the standard pattern:

• personal form of musieć
• plus infinitive

Because standard Polish often changes the direct object to the genitive when the verb is negated. This is called the genitive after negation.

Compare:

• Muszę piec ciasto. = I have to bake a cake / the cake
• Nie muszę piec ciasta. = I don’t have to bake a cake / cake

Here:

• ciasto is the non-negated form you would expect as the object
• ciasta is the genitive form used under negation

This is a very important grammar pattern in Polish, and it also appears when the object belongs to an infinitive after a negated verb like nie muszę.

Yes, upiec is possible, but the nuance changes.

• piec is imperfective: it presents the action more generally, as an activity
• upiec is perfective: it focuses on completing the action and getting the result

So:

• Nie muszę dziś piec ciasta = I don’t need to do any cake-baking today
• Nie muszę dziś upiec ciasta = I don’t need to get a cake baked today / finish baking one today

In your sentence, piec sounds very natural because the speaker is talking about whether the task is necessary at all, not about completing it by a certain point.

Więc means so, therefore, or thus.

It connects the two parts of the sentence:

• there is still chocolate pudding in the fridge
• so I don’t have to bake a cake today

It is a very common everyday linking word. It sounds natural and conversational here.

Because więc is a coordinating conjunction linking two clauses, and in Polish it is normally preceded by a comma.

So:

• W lodówce jest jeszcze budyń czekoladowy, więc nie muszę dziś piec ciasta.

That comma is standard Polish punctuation.

They mean the same thing: today.

• dziś = shorter form
• dzisiaj = longer form

Both are normal and very common. In this sentence, dziś is just a compact, natural choice.