Dzisiaj jest mi zimno, chociaż w domu jest ciepło.

Polish uses jest mi zimno to mean I feel cold / I am cold (as a sensation).

• jest mi zimno is an impersonal construction: literally “it is cold to me”.
• jestem zimny literally means “I am cold (to the touch)” – like a cold object or a corpse. It describes your physical temperature as an object, not your feeling.

So:

• Dzisiaj jest mi zimno. = Today I feel cold.
• On jest zimny. = He is cold (his body/object is cold; or metaphorically, he is cold as a person).

Mi is the dative form of ja (I). It corresponds to “to me”.

• Case: dative (komu? czemu? – “to whom? to what?”).
• Construction: jest zimno komuś = “it is cold to someone”.

So:

• jest mi zimno = it is cold to me
• jest ci zimno = it is cold to you (singular, informal)
• jest mu zimno = it is cold to him

Polish often uses the dative to show who is experiencing a state or feeling in these impersonal sentences (zimno, gorąco, smutno, wesoło, etc.).

Both mi and mnie are dative forms of ja, but they are used slightly differently.

• mi is the usual, weak/clitic form, very common in speech:
  
  • Dzisiaj jest mi zimno. – the most natural version.
• mnie is the stressed/emphatic form, used for contrast or strong emphasis:
  
  • Mnie jest zimno, a jemu nie. – I am cold, but he isn’t.

So you normally say jest mi zimno, and use mnie only if you really want to stress me.

In jest mi zimno and w domu jest ciepło, zimno and ciepło are adverbial (neuter) forms of adjectives used in an impersonal way.

• Base adjectives: zimny (cold), ciepły (warm).
• Adverb/impersonal forms: zimno, ciepło.

They function like English “it is cold/warm”:

• Na dworze jest zimno. – It’s cold outside.
• W domu jest ciepło. – It’s warm at home.

They do not agree in gender with the person; they are just general descriptions of the temperature.

Polish often drops jest in the 3rd person present (on/ona/ono):

• W domu ciepło. instead of W domu jest ciepło.
• Na zewnątrz zimno. instead of Na zewnątrz jest zimno.

In your sentence:

• Dzisiaj jest mi zimno, chociaż w domu jest ciepło.

You can say more colloquially:

• Dzisiaj mi zimno, chociaż w domu ciepło.

All of these are correct; including jest sounds a bit more neutral/standard, while dropping it can sound more spoken and elliptical. It’s less common to drop jest before mi, but Dzisiaj mi zimno is still fine.

Yes, Polish word order is flexible, and different orders change the focus/emphasis, not the basic meaning.

Examples:

• Dzisiaj jest mi zimno. – neutral: “Today I feel cold.”
• Mi jest dzisiaj zimno. – emphasizes me: I’m the one who is cold (maybe others are not).
• Dzisiaj mi jest zimno. – emphasizes today vs other days.

For the second part:

• W domu jest ciepło. – neutral.
• Ciepło jest w domu. – contrastive: it’s at home that it’s warm (maybe not outside).
• W domu ciepło. – shortened, typical spoken style.

All are grammatical; context decides which sounds most natural.

Chociaż means “although / even though” and introduces a contrastive subordinate clause.

• Dzisiaj jest mi zimno, chociaż w domu jest ciepło.
  = Today I feel cold, although it is warm at home.

Differences:

• ale = but; it usually starts a new main clause:
  
  • Dzisiaj jest mi zimno, ale w domu jest ciepło.
    (Two separate facts: I’m cold, but the house is warm.)
• chociaż / choć = although; it introduces a dependent clause explaining the surprising contrast.
• mimo że = despite the fact that; similar to chociaż, but sounds a bit stronger/more explicit:
  
  • Dzisiaj jest mi zimno, mimo że w domu jest ciepło.

In everyday speech chociaż and choć are very common; mimo że feels slightly more formal or emphatic.

In Polish, you normally place a comma before conjunctions that introduce a subordinate clause, such as chociaż, że, ponieważ, kiedy.

• ..., chociaż w domu jest ciepło.

Here “chociaż w domu jest ciepło” is a dependent (subordinate) clause explaining the contrast, so it must be separated by a comma from the main clause “Dzisiaj jest mi zimno”.

Yes, you can:

• Dzisiaj jest mi zimno, choć w domu jest ciepło.

Choć is essentially a shorter, slightly more colloquial form of chociaż in this meaning (although). In this sentence, they are interchangeable and the meaning remains the same.

There is also a different use of choć meaning “at least / even / come on” in some contexts, but here it’s just “although”.

Dzisiaj means “today”. Dziś is a shorter, slightly more formal/literary but very common variant; they are interchangeable in meaning.

You can say:

• Dzisiaj jest mi zimno, chociaż w domu jest ciepło.
• Dziś jest mi zimno, chociaż w domu jest ciepło.

Placement is flexible:

• Dzisiaj jest mi zimno... (most neutral)
• Jest mi dzisiaj zimno... (focus on today)
• Jest mi zimno dzisiaj... (often contrast with other days: “I’m cold today, unlike before.”)

All are grammatical; the difference is mostly emphasis and rhythm.

W domu = “in/at home”, using:

• Preposition w (in),
• plus the locative case of dom: dom → w domu.

Contrast:

• w domu – in/at home (location, where?):
  
  • W domu jest ciepło. – It’s warm at home.
• do domu – to home (direction, where to?):
  
  • Idę do domu. – I’m going home.
• z domu – from home (from where?):
  
  • Wyszedłem z domu. – I left home.

So in your sentence we talk about the state of temperature at a location, so w domu (locative) is required.

To negate this kind of sentence, nie goes before the verb jest:

• Dzisiaj nie jest mi zimno, chociaż w domu jest ciepło.
  = Today I am not cold, although it is warm at home.

You cannot say:

• *Dzisiaj jest nie mi zimno – incorrect.
• *Dzisiaj jest mi nie zimno – also incorrect.

If you drop jest, you usually keep it positive; the negative version sounds better with jest present:

• Positive spoken: Dzisiaj mi zimno.
• Negative: Dzisiaj nie jest mi zimno.

Approximate pronunciations:

• dzisiaj: /ˈd͡ʑi.ʂaj/ (in careful speech) or /ˈd͡ʑi.saj/

  • dź / d͡ʑ – like “j” in “jeans”, but softer.
  • si before a vowel becomes a soft ś sound.
    So it’s roughly like “JEE-shay” or “JEE-say” (but with a softer sh/s).

• chociaż: /ˈxɔ.t͡ɕaʂ/

  • ch – like German “Bach” (voiceless velar fricative).
  • ci / ć / t͡ɕ – a soft “tch” sound, like a very soft “ch” in “cheeks”.
  • Final -sz / -ś / -ż / -cz-type sounds are pronounced clearly; -aż here is like “ash” but with the sh further back.

Very rough English approximations:

• dzisiaj ≈ “JEE-shy”
• chociaż ≈ “HO-tyash” (with the h like in “Bach”, not English “home”).