Ich bin kalt is a grammatical German sentence, and it is not about the room. It is about the kind of person you are, and it says so plainly: cold, unmoved, hard to reach. The next attempt reaches for the haben that works in ich habe Hunger, and ich habe kalt is not a sentence in standard German. What German says when the room is cold puts you in the dative and leaves the subject slot empty: mir ist kalt.
A set of states works that way. The person goes into the dative and the sentence has no personal subject: mir ist kalt, mir ist warm, mir ist schlecht, mir ist langweilig. You are not doing the being in any of them. The state is reported, and the dative pronoun says who it reached.
Alongside it, a set of things you carry takes haben and a noun: ich habe Hunger, ich habe Durst, ich habe Angst. Here you are the subject again, and what English keeps as an adjective is a noun in German, which is why ich habe hungrig is not available either. The noun is the whole construction, and it is why these three are learned as phrases rather than assembled.
The same dative person turns up with a family of verbs that describe how something lands on you: das gefällt mir, mir tut der Kopf weh, mir fehlt ein Wort. Each of those does have a subject, and it is never the person: it is the thing, the head, the word that is missing. Italian builds one of its most ordinary sentences on that shape, and the post on mi piace against mi piacciono is about what the shape does to the verb there.
Mir ist kalt.
Ich bin kalt.
Ich habe Hunger.
Das gefällt mir.
Mir tut der Kopf weh.
Mir is doing one job across all of those, and it is never the subject. What varies is whether the sentence has a subject at all. The cold has none, so nothing in mir ist kalt is nominative. The head is one, and it brings its own article with it, which is why der Kopf is enough: German has already said whose head it is, in the dative, at the front. Mir tut mein Kopf weh and mein Kopf tut weh are both heard as well, with the possessive doing the work the article was doing.
The edge that trips people up
This is a list of words rather than a rule about inner states, and that has to be said in the same sentence that gives the list, because the list invites a generalisation it will not carry. Ich bin müde is the ordinary way to say you are tired. Ich bin traurig is the ordinary way to say you are sad. Both put you in the subject slot with an adjective after bin, exactly where English puts you, and a learner who has decided that feelings go into the dative will produce German that nobody says. Cold, warmth, sickness, boredom and dizziness take the dative (mir ist kalt, mir ist warm, mir ist schlecht, mir ist langweilig, mir ist schwindelig); tiredness and sadness do not. There is no test underneath that, only the words as you meet them.
Mir ist schlecht, aber ich bin nicht müde.
German also has a plain verb for being cold, ich friere, with you as the subject and nothing in the dative anywhere. It stands next to mir ist kalt.
The front of the sentence takes two fillers. Mir ist kalt and es ist mir kalt both exist, because that first slot takes either the dative pronoun or the placeholder es, and neither is being ranked here. The dative pronoun in front is the convention of this pattern, and a convention is all it is. Either way the verb keeps second place, the way a German verb does in any statement.
Mir ist kalt in dieser Wohnung.
How to hear it
Decide what the sentence is about before you choose the verb. A state that arrives, a temperature, a mood, a pain, puts you in the dative and leaves the sentence with no personal subject. A state you carry with you takes haben and a noun after it. And when you mean what kind of person you are, ich bin is right, and it says exactly that.
What makes the first of the three hard to reach in time is that English gives you nothing to convert. “I am cold” has a subject, a verb and an adjective, all three in the order German will not use, so the German sentence has to be available as a phrase before you need it rather than built while you need it. Mir ist kalt is quicker to have ready than to work out.
The dative verbs pay the phrase back, because once mir is in place as the person a thing happens to, das gefällt mir, mir fehlt ein Wort and mir tut der Kopf weh are the same move with a subject added, and a good deal of ordinary German is made of them.
There is no ich in mir ist kalt. The German sentence for being cold does not have you as its subject at all: it states that it is cold and names the person the cold reached. Ich habe Hunger puts you back at the front with something to carry. Ich bin müde gives you the subject and the adjective English would expect. Which of those shapes a state takes is something the state decides, and you learn it with the word.
