Yes, if you had income reported to you on Form 1099-NEC, that generally means that the company who paid you considered you an independent contractor. As such, you are self-employed for the work that you did for that company and you would report your income and expenses on Schedule C.
The question about whether you are required to file a Form 1099 for anyone that you paid over $600 is asking whether your Schedule C business hired another independent contractor to do work for your company and you need to issue a Form 1099 for any payment you made to them. Typically, the answer would be no unless you pay sub-contractors through your business.