Learn Russian: Мой нос холодный.

In Russian, the present tense of the verb "to be" (is, am, are) is normally left out. You just put the subject (Мой нос) right next to the adjective describing it (холодный).

Why does холодный end in -ый here?

Холодный is an adjective describing нос (nose). Because нос is a masculine noun and is the subject of the sentence (in the nominative case), the adjective must take the standard masculine ending -ый.

To say "I am cold," I learned to say Мне холодно. Why don't we use холодно in this sentence?

Мне холодно uses the adverb холодно to describe a general feeling or state (literally "To me it is coldly"). However, in this sentence, you are describing a specific physical object (your nose). When describing a noun, you must use an adjective like холодный.

Do I always need to use Мой (my) when talking about body parts?

Not always. In Russian, possessive pronouns like мой are often dropped when talking about body parts if it is obvious whose body it is. But keeping it here is perfectly fine and sounds natural, especially if you want to emphasize that it is your nose specifically.