Maybe this will help, below is a pdf version of what your teachers have to fill out online: https://www.commonapp.org/CommonApp/...1_download.pdf
Counselors have to submit the school report,mid year report, and the final report: https://www.commonapp.org/CommonApp/...t_download.pdf https://www.commonapp.org/CommonApp/...r_download.pdf https://www.commonapp.org/CommonApp/...t_download.pdf
When you ask your teacher for a recommendation (and the school you are applying to uses the common app) assuming your teacher says yes, you have to go to the school forms tab in your common app account and add school official; here it will ask you for your teachers email address, name and title. When you click the submit button your teacher will get an email saying that you requested them to fill out the recommendation forms for you they have three options: they can deny your request, say that they will do it on paper or accept your request and complete it online (most teachers and counselors do it online). If it's their first time on the common app for this application season they will also get emails with their username and password. Keep in mind that the moment you send the email the rest is up to your teacher and, assuming you waived your rights to see the recommendation letter, you don't get to see what your teacher writes. Essentially the counselor's rec works the same way the only thing that changes is the amount of forms they have to fill out, but on your end it's the same.
As for your final question, it really depends on the university your applying to. Typically it's 1-2 teacher recs the counselor rec and sometimes some other universities require others you have to read on their website to be sure.