Learn French: Elle choisit une chaussure rose. - She is choosing a pink shoe.

Choisir is a regular -ir verb. To conjugate it for il, elle, or on, you drop the -ir from the end to get the stem (chois-), and then add the ending -it. This gives us elle choisit.

The translation says "is choosing", but doesn't elle choisit mean "she chooses"?

It means both! French does not have a separate "is doing" tense like English does. The present tense in French (elle choisit) translates to both "she chooses" and "she is choosing", depending on what makes sense in the context.

Why does rose come after chaussure?

In English, we put the color before the noun ("pink shoe"). In French, colors and most other adjectives come right after the noun they describe, making it chaussure rose (literally "shoe pink").

Chaussure is feminine. Do we need to add an extra -e to rose to make it agree?

No extra -e is needed here. When an adjective like rose already ends in an unaccented -e in its default dictionary form, it stays exactly the same for feminine nouns.