Can single mother and a child live in a rented apartment?

It really depends.  Assuming you have the correct legal status in Saudi, inside the compounds, no issue.  But some compounds may not have single accommodation and may turn you down because of that.

The problem comes outside the compounds as there are not that many female expats working in the Kingdom outside of the professions of nursing, medical staff, teachers, flight attendants etc.   Almost all are provided separate female accommodation by their companies in dedicated buildings / compounds e.g. Saudi cabin crew stay in a dedicated female only compound.  So landlords don't see a lot of requests from single females and as such are very reluctant to rent to them.  Plus depending on the area, the other tenants might have issues.   In your situation, I would advise to check with compounds only.   The other alternative is to look for serviced hotel apartments.

I think if you've a working permit you can rent an apartment

Yes you can easily rent a Flat, Villa or anything. Just with a Valid Iqama.

Yes if iqama is valid and divorce doc are available

Yes, you can, I can help you.

it not prohibited!! . you can live where ever you want in Saudi Arabia. but you should have Iqama valid