Learn Finnish: Hymyileekö nainen? - Is the woman smiling?

This is vowel harmony in action! Look at the vowels in the verb hymyilee. It contains y, i, and e. Because y is a front vowel (and i and e are neutral), the suffix must also use a front vowel. That's why we use -kö.

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

Finnish doesn't use a separate word for "is" when talking about an ongoing action. The single verb hymyilee translates to both "she smiles" and "she is smiling". By attaching the -kö question ending to it and moving it to the front, the whole word becomes "Is she smiling?".

The dictionary word for "to smile" is hymyillä. Why does it change to hymyilee here?

Hymyillä belongs to a group of verbs where the stem changes before you can add most endings. The stem becomes hymyile-. To make the 3rd person singular ("he/she/it") form, you double the last vowel of the stem, giving you hymyilee (she smiles).

Could I attach the question ending to the woman instead, like Nainenko hymyilee?

You could, but it changes the meaning! The -ko or -kö ending attaches to the specific detail you are asking about, and that word usually jumps to the front of the sentence. Hymyileekö nainen? asks "Is the woman smiling?". Nainenko hymyilee? asks "Is it the woman who is smiling?".