Client want's to be able to answer Yes to a question in a custom list that will cause three of the fields to populate in another site in an announcements web part. Is this possible through workflow?
Based on what I think you're getting at, you can use a "Set a Condition" action to evaluate whether your boolean is a yes, if it is then using the "Update Multiple Items" action you can update an item in a different list on a different site with the values that you define.
Once you update the Announcements list this, of course, will be reflected in your announcements web part.
Yes that is possible. You can use a workflow task action to perform that and modify the outcomes to be yes/no.
Place a workflow on the item in the custom list and set it to run when an item is modified. Set the condition or use run if at the beginning to evaluate if the field is yes/no. If yes, use the create item to generate an item in the announcements list with the correct metadata and it should show up in the webpart once completed. I attached a starter workflow for you.