Learn Russian: Наш врач очень молодой.

The Dative case is used when stating exactly how old someone is with a number (literally "to him are 20 years"). However, when you are simply describing someone with an adjective like молодой (young) or старый (old), you just use the standard Nominative case, exactly like you do in English.

Where is the word for "is" in this sentence?

In Russian, the present tense of the verb "to be" is almost always skipped. The sentence literally translates to "Our doctor very young." You just pause slightly where "is" would normally be.

If our doctor is a woman, do we still use the masculine words наш and молодой?

The word врач is grammatically masculine, so words directly attached to it usually stay masculine, like наш врач (our doctor). However, when describing the female doctor at the end of the sentence, it is very common and natural to switch to the feminine adjective молодая.

Why does the adjective молодой end in -ой instead of the usual -ый?

Most masculine adjectives end in -ый (like старый - old) or -ий. But whenever the spoken stress falls on that final ending, Russian spells it as -ой. In молодой, the heavy emphasis is on the very last syllable (ma-la-DOY), so it gets the -ой ending.