The day after that 60th day (assuming its a weekday) you can file the unlawful detainer in court.

On those three occasions the tenants unlawfully detained until the sheriff came, but were out in less than 30 days.

If the rent is not paid in full by the 10th, I proceed with unlawful detainer action.

Seems like the unlawful detainer filings don't become public until 60 days after being filed and even then you have to search by the first AND last name of the TENANT or by case number.

I went to the court and filed an unlawful detainer on Friday and am currently trying to locate my tenant so I can have her served.

Without a legitimate license, my understanding is that you will have a difficult time getting through adverse situations as a landlord such as evictions/ejectment because you are essentially running an unlawful operation. 

Then you'll have to proceed with evicting the tenant (i.e. unlawful detainer lawsuit).

If they don't move, file your unlawful detainer and have them served.  

After the 3 day notice expires you have to file an "unlawful detainer", the legal term for an eviction.

Public information in the superior court indexes for California unlawful detainers are not available until 60 days after the filing (per state code).