My Form I-20 has the following written on the third page:
To transfer schools, first notify the DSO at the school you are attending of your plan to transfer, then obtain a Form 1-20 from the DSO at the school you plan to attend. Return the Form 1-20 for the new school to the DSO at that school within 15 days after beginning attendance at the new school. The DSO will then report the transfer to the Department of Homeland Security (DHS). You must enroll in the new school at the next session start date. The DSO at the new school must update your registration in SEVIS.
Here, DSO means "Designated School Official".
Moreover, the I-20 also has the following mentioned:
Report to School Named on Your Form I-20 and Visa:
Upon your first entry to the United States, you must report to the DSO at the school named on your Form 1-20 and your F-1 visa (unless you are exempt from visa requirements). If you decide to attend another school before you enter the United States, you must present a Form 1-20 from the new school to a U.S. consular officer for a new F-1 visa that names the new school. Failure to enroll in the school, by the program start date on your Form 1-20 may result in the loss of your student status and subject you to deportation.
This clarifies my first question. As for my second question, I did not get a B1 interview appointment, so I anyways cannot enter the United States more than a month before my course starts. If you are already in the US when you get your I-20, you can apply to change the visa class. If not I think you have to reenter, but I'm not sure. I will appreciate input from someone else regarding that.