In CALIFORNIA, does anyone know if there is a maximum amount one can be charged a late fee if rent is not paid on time for that month?
Is it legal to charge someone an amount each day for a late fee until rent is paid?
Also, I was told if you pay at least half of the rent and landlord excepts the money towards the rent than he can't charge a late fee. Is this accurate?

Well you will find some interesting arguments here:

California Tenant Law - Free legal advice for California renters' rights

But frankly I don't agree with a lot of his premise. I think the key is what is written into the lease. If the lease specifies a late charge and the amount of the charge and you signed the lease with that clause, then I think a court will uphold it. Unless it is way out of line.

And no, rent is not paid until the full amount is received. So until the full amount is received the rent is considered late and subject to any penalty written into the lease.