Learn German: Meine Großmutter kocht sehr gut. - My grandmother cooks very well.

Because Großmutter is a feminine noun, and it is the subject of the sentence (in the nominative case). In German, possessive words like "my" have to match the gender of the noun they belong to. For feminine nouns in this case, you add an -e to the end, making it meine.

gut usually means "good". Why is it translated as "well" here?

In English, we have to change the word depending on what we are describing: we use "good" for nouns (a good cook) and "well" for actions (cooks well). In German, it is much easier! You don't need a separate word or an "-ly" ending. The exact same word, gut, is used for both.

If I want to say "My grandmother is cooking very well" right now, does the verb change?

No, it stays exactly the same. German does not have an "is doing" (present continuous) tense. The simple present verb kocht covers both "cooks" as a general habit and "is cooking" right at this moment. If you really want to emphasize that it is happening right now, you could just add a time word like gerade (currently) after the verb.