Learn Russian: Осторожно! Здесь дверь.

Yes! Just like хорошо (well) and плохо (badly), осторожно is an adverb. It literally means "carefully" or "cautiously." When shouted on its own, it functions just like the English warning "Careful!" or "Watch out!"

Why isn't there a word for "is" in the second sentence?

In Russian, the present tense of the verb "to be" (am, is, are) is almost always skipped. So Здесь дверь literally translates to "Here door," but it means "Here is a door" or "There is a door here."

Where does the "There" come from in the translation "There is a door here"?

English often uses "there is" or "there are" to point out that something exists. Russian doesn't use a filler word like "there" for this. It simply states the location (здесь) and the object (дверь).

How do I know the gender of дверь since it ends in a soft sign (-ь)?

Дверь is a feminine noun. Nouns ending in a soft sign can be either masculine or feminine, so you just have to memorize the gender when you learn the word. A helpful hint: many everyday household objects ending in a soft sign, like дверь (door) and кровать (bed), are feminine.