Learn Russian: Юность — это короткое время.

In Russian, the present tense of "to be" (am/is/are) is usually dropped. When you are defining something or saying "Noun A is Noun B", Russian often uses a dash followed by это (this/it) to link the two ideas. Literally, this reads like "Youth — this [is a] short time."

Since юность ends in a soft sign (-ь), how can we tell its gender?

The word юность is feminine. A great tip to remember: almost all Russian abstract nouns ending in -ость (like новость for news, or молодость for youth) are feminine.

Why does the adjective короткое have a neuter ending (-ое) if время ends in -я, which is usually feminine?

You are right that -я is usually a feminine ending, but время is a special exception. It belongs to a small group of nouns ending in -мя that are actually neuter. Because время is neuter, the adjective must match it with the neuter ending -ое.

Since we are learning about expressing age in the past, how would I say "in my youth" instead of using the "Dative + было + years" structure?

You would use the Prepositional case: в юности. It is a great alternative to the structure you are currently learning. Instead of specifying your exact age, you can simply start your sentence with В юности... (In my youth...).