If you are already commissioned, in my experience you write your own Letter and route it up for them to tweak and sign. If you are talking about civilian bosses, then yea they should write it, but I would provide a list of your accomplishments and whatnot. I would ask for the letter ASAP so you dont have to worry about them.
For the AF, the format is typically a paragraph about your extracurricular (i.e volunteering), a paragraph about the work youve done for the person signing the letter, a paragraph about your past achievements in the AF (awards, strong impacting stuff, etc) and then the last paragraph about how you are a hard worker and a perfect fit for whatever you are applying for. I think the Tongue and Quill has a template
Yes they should sign.