- mir" is Dativ, "mich" is Akkusativ. Some verbs are followed by Dativ (indirect object), some by Akkusativ (direct object). For example, ich danke dir (the verb "danken" takes the Dativ) but ich liebe dich ("lieben" is followed by Akkusativ). When you learn new verbs you also need to learn whether they take the Dativ or Akkusativ or both.
- "Mir ist kalt", means I feel cold. "Ich bin kalt" is also correct but refers to your personality, as in I am a cold (hearted) person. Hope this helps. If you need further help with your German please take a look at my website https://ninabuschmann.com