1. From your description, the name of your institution has been legally approved by the local administrative department of education, and there is no malicious unauthorized use of other people's enterprise names, which leads people to mistake them for other people's goods. Should not be considered as infringement.
2. Article 66 of the Law on the Promotion of Private Education stipulates: "Measures for the administration of operating private training institutions registered with the administrative department for industry and commerce shall be formulated separately by the State Council." Although the State Council has not issued relevant regulations so far. However, at present, for-profit institutions in private education are registered with the industrial and commercial administration departments in accordance with the Implementation Measures for the Administration of Enterprise Name Registration.
3. If the other party is a well-known trademark, you have to use and register the other party's trademark to infringe (note that trademark and enterprise name are two different things). In accordance with the relevant provisions of the Law of the People's Republic of China against Unfair Competition and the Trademark Law. If you use the other party's trademark without registration, it is definitely infringement.
if a trademark that is the same as a well-known trademark is registered, the trademark owner or interested party may request the Trademark Review and Adjudication Board to cancel the registered trademark within five years from the date of trademark registration. For malicious registration, the owner of a well-known trademark is not subject to a five-year time limit. Therefore, if you register without malice, the other party should put forward it within 5 years.
4. The management of domain names is based on the "Measures for the Administration of Domain Names in China", and the domain name registration service follows the principle of "first application, first registration". If it is not illegal to rob the country, the country and the public, the domain name registration authority can protect some reserved words and pay the fees on time, and the other party has no legal reason to claim your domain name.
5. As far as your situation is concerned, both your academic sample and the other school may not be registered in the industrial and commercial department. Because the premise of such regulations as trademarks and names is legal registration, it is suggested that you register the school in the name of the company first, and it is more beneficial to protect your rights to apply for trademark registration with relevant name marks and logos.