Let me start by saying I'm not a SZ. I have SZ. Symantics I know.
To answer your question...yes! People with SZ can and do live a "normal" life. Speaking from my own life I own my house and two cars. I pay my bills on time. I don't have huge credit card debts, because I pay them off every month. I worked long enough to get a disability pension. I am married. Have 2 children and 2 grandchildren. I work around the house. I cook and clean. I shop. I bathe regularly. I take my meds and see my doc on a regular basis. I travel.
I do all that even though I hear voices (which is everyday for the last 29 years).
I don't know if I'm normal, and I don't know what you would call "normal" but I have made a good life for myself and wife.
Peace,
Dave