Yes you will need to complete a FAFSA in order for your kid to get the federally funded loan. You can complete it any time you choose during the academic year. The early filer thing is for limited funding aid types like federal work study and SEOG.
Need aware schools consider your ability to pay at some point during the admissions process. If your financial aid forms show you as full pay, this isn’t going to affect admissions because the school doesn’t need to give you aid.
You can submit the FAFSA to any schools you want to…and you can easily add schools later if you so choose.
This is college dependent. You will need to check each college for their policy. My kids both went to colleges which, at the time, required these forms to be completed even for merit awards. But anyway…you need to check each college.
Your kid will be eligible for a $5500 Direct Loan if you complete the FAFSA. You do not have to complete the Profile to get this federally funded loan.