Ok. Let me bring some closure to this, as Headmaster and SU Bear are pretty much saying the same thing and both are correct.
In NY, public school districts are required to have a Code of Conduct, which is a Board of Education approved and adopted policy that, among other things, sets forth the rules of conduct for students and visitors on school district property. It is required to be reviewed annually, and any amendments generally require a public hearing before the Board takes action. This doesnt always happen, and usually, nobody cares. Generally, there are due process protections mandated by the law that must be provided before a student can be suspended from school, and the protections grow to entitlement to counsel and full evidentiary hearing for suspensions beyond five days (other protections exist for students w/ disabilities). The District bears the burden of proof, but the evidentiary standard is quite low ("more likely than not"). Student's do have the right of confrontation, which can muddle things when the only witnesses are students.
School districts cannot discipline (out of school suspension) for incidents that do not have a "nexus" to school. The "nexus" standard allows that students may be disciplined for conduct that occurred outside of the school that may endanger the health or safety or pupils within the educational system or adversely affect the educative process. Creative schools can usually find a way to meet this standard when they deem it necessary. However, for some kid who commits an act, completely outside the district, involving people who also reside outside the district, during a time when school is not in session, the nexus will be very tenuous at best.
School districts are not required to have a separate athletic code of conduct, also adopted by the board, but every district I know of has an athletic code. They first came in vogue in the late 90s and were premised around the notion that playing sports for a school is a privilege, these student-athletes are representing the school community, and therefore should be held to a higher standard. Accordingly, students can be "suspended" from athletics for a broader spectrum of reasons, and often a nexus to school is not necessary. The school district really makes the rules via its policy, and then has to enforce its own policy. Schools have actually been tightening these policies because with the rise of social media, kids off-campus behavior is much easier to monitor. Usually, an arrest will violate an athletic code of conduct, due to the discredit it brings the school district.
Schools are only required to educate students who are considered to be of compulsory education age (i.e., through the school year they turn 16). Students with IEPs are required to be educated and receive their special ed supports up to the age of 21. Most schools will educate students on suspension even beyond compulsory education age because it is in everyone's best interest for those students to earn credits and graduate.