Learn German: Meine Mama kocht sehr gut.

In German, the same word is usually used for both adjectives (describing a noun, like "a good soup") and adverbs (describing an action, like "cooks well"). You don't have to learn a separate word like you do in English; gut covers both "good" and "well".

Why is it "Meine" with an "-e" at the end, instead of just "Mein"?

The word Mama is a feminine noun. When a feminine noun is the subject of the sentence (in the nominative case), possessive words like "my" need an -e added to the end, making it Meine.

Could this sentence also mean "My mum is cooking very well" right now?

Yes! German doesn't have a separate "is doing" tense (present continuous). Meine Mama kocht can translate to "My mum cooks" (as a general fact) or "My mum is cooking" (happening right now), depending on the context.

Is "Mama" used by adults in German, or is it mostly a word for children?

Adults use Mama (mum/mom) and Papa (dad) all the time in everyday spoken German when referring to their own parents. It is affectionate but completely standard, whereas Mutter (mother) can sound a bit more formal or distant.