Education: Graduate student summer internship tax treatment

I am a graduate student who is filing taxes for a summer internship I performed in Summer 2024, and I have a question on the payment received by the company, as I received a 1099-NEC form. See below:

• Situation: This was a third-party startup company internship that was available through and partially subsidized by my school (stipend program for a summer internship experience if a company does not have the funds to fully compensate them, the school will provide a stipend / scholarship, so the total compensation is a certain amount). The school paid for a majority amount of $X (which was not part of the 1098-T, but I included under "Education - Other Scholarships/Grants/Fellowships", no questions here) and the third-party company paid for a portion $Y. I received a 1099-NEC, which is for independent contractors.
  
  • Education component: I am a fully enrolled graduate student who takes classes full time throughout the year, with the exception of Summer 2024 for the internship. I am required to complete a 6–12-week full time summer internship as part of the work experience degree requirement and this work experience is a graduation requirement. Does this mean I should be exempt from self-employment FICA tax ("performs services as an incident to and for the purpose of pursuing a course of study")? 
  • Employee vs. independent contractor component:
    • Behavioral:
      
      • Type and degree of instructions given: Had weekly check-ins to discuss work done and top priorities for the week; degree of instruction depended on task - some had little (e.g. email and reach out to X potential partners) and some had more direction (e.g. brainstorming together new initiative)
      • Evaluation systems & training: Had an initial onboarding, then subsequent weekly check-ins and feedback on work done over the internship period
    • Financial:
      
      • Did not have significant investment, unreimbursed expenses, or opportunity for profit and loss (used personal laptop, occasional commute expenses)
      • Services available to the market: I did not advertise, have a visible business location, or would work in this role otherwise (have not previously worked directly in this type of role). If I had not done this internship, I would have had to find another summer internship for work experience that may or may not have been similar to this role
      • Method of payment was bi-weekly via bank transfer, totaling the flat $Y amount over the internship period
    • Type of relationship:
      • Written contract: I received an offer letter that explicitly states I have a full-time internship position with a start and end date who I report to and my responsibilities
      • No employee benefits
      • No expectation that the internship/work would extend beyond project time to a full time role
      • Services provided as key activity: My work was presented as the company's own and I do not control or direct it
• Questions (and would appreciate any additional IRS guidance / verbiage for clarity if possible so I can double check that I am reporting correctly): 
  • Should the third-party portion of the stipend be considered nonemployee compensation (e.g.  employee vs. contractor)?
    • If not, what form should I have gotten / how should this be reported (e.g. within Education - Other Scholarships/Grants/Fellowships)?
    • If the 1099-NEC is correct, how do I fill out the Schedule C as I technically was not operating a business or a sole proprietor practitioner? It requires me to fill out business information, code, accounting method, whether I issued payments requiring a 1099, etc.