Do the teachers write different letters and need different forms for each college/university?
As a rule, they do not. The teachers will write one letter that can be sent to multiple colleges (e.g. it will not recommend you to a specific college; just college in general). Occasionally, teachers will need to fill out different recommendation forms. If the college to which you are applying does not accept the Common Application, for example, and yet they still want a teacher letter and form, you will need to complete the top part of that college’s own form, sign it, and give it to the teacher(s).