Questions about trust conversion (on death of trustor)

[Daniel Miller]

I am beneficiary on my father's living trust.  We are in California.

He just passed away today, so I'm beginning the process of transferring assets from his trust to my accounts (which is what the trust says I should do, since I am also successor trustee on that trust), once I get death certificates in hand. 

However, I'm unclear on exactly what I should with his house?
> do I just fill out some change-of-ownership form with the county?
> Are there any federal or IRS documents that I need to fill out for this process?
> I believe that when the property changes hands, that the 'cost basis' changes to its current market value;
do I understand this correctly?  Do I need to get the house re-appraised for this purpose?  Do I need to communicate anything to the county about this?
> I've been renting out his house for a number of years, to help pay his assisted-living costs;
y'all have helped me with that on a couple of occasions!  At tax time, I've been applying depreciation to the house; how does this affect my actions or calculations now?

And if there is somewhere else that is more appropriate for these questions, feel free to point me!
Thank you again for all your help over the years.

Dan Miller

Fremont, CA