I am currently caring for my partner with Alzheimer’s. Could I apply to be a professional caregiver and continue to care for him?

Never heard of applying to be a professional caregiver. In my State a Certified Nurses aide takes a course that lasts about 10 weeks. Then the students take a test and if pass are certified by the Nursing Board. The only way I see you getting paid is thru Medicaid. So check with them to see what programs are available for Caregivers. You will probably only be paid minimum wage and not for 8 hrs a day.

That depends on whether he can QUALIFY as needing care. First step.
Then whether or not your state has Medicaid or other programs that encourage keeping a loved one in the home and cared for by friend or family.
That is something for you to research for your own state and I wish you good luck.
If it proves impossible an Elder Law Attorney can assist you, as well as a social worker.

Pretty sure that you can although the laws very state by state, Your partner would become your employer and pay income taxes on you.