Part 91 operations, unless the school has a Part 141 certificate, or uses the aircraft for charter work, which then incorporates Part 135. Maintenance is done under Part 43, unless the school has a repair station certificate, in which case it incorporates Part 145.
TCDS is type certificate data sheet; best considered the pedigree or birth certificate for a line of engines, aircraft, etc. It spells out the certification limitations and details (parameters) for a given piece of equipment. It is what the FAA issues when the aircraft is given it's blessing to press forward: the aircraft is recognized as a type, and the TCDS holder becomes the manufacturer. The Type Certificate Data Sheet is regulatory and conformity to the TCDS is required, unless authorization is given to deviate (eg, STC, or Supplemental Type Certificate), or through the use of a CDL (configuration deviation list) or MEL (minimum equipment list) or field approval (form 337) or properly altered by 91.213.
The certification regulation for the Cessna 172 is 14 CFR Part 23, which will be detailed in the type certificate data sheet.
https://www.ecfr.gov/cgi-bin/text-idx?S ... 2_02.tpl#0