You're following the correct procedure. A Customer Quotation (or any other Transaction type) has "Generate PDF" and "Send Email" buttons at the top of the screen. These buttons become split buttons (drop-down with multiple options) if multiple output document (PDF) templates are available (for that object type).
As you've already identified, multiple email templates are not available on the Customer Quotation screen. You select "Send Email", which attaches the PDF to the email, then you choose the email template (as long as it's activated and for that object type)
You then add the recipients. Bear in mind the F.A.O. field on the Quotation Screen is free text. It doesn't "look-up" or link to a person record's email address, so it can't be automatically added.
Please feel free to add an idea to the Workbooks Enhancements & Suggestions Forum if you'd like this, or a similar feature, to be considered for future development. Thank you.