if a thirteen year old's friend commits suicide, when/do they get to read their suicide note?

Not everyone leaves a suicide note. I am sorry that you have to go through such a sad situation and wish you well on your path to recovering from this.

They don't unless the family allows it.

The police will read it when the body is discovered. Depending on its contents, they might share it with the family.

Once all investigation are finished, the death enquiry rules it's suitable then the note may be passed to the next of kin