Legal Question in Wills and Trusts in California

The settlor can select anyone they want and give them whatever powers, within reasonable limits, they want. Anyone includes the settlor themself.

Yes, the settlor can have someone else be the trustee and that person will have full trustee powers.

In larger wealth management family trusts of high-net-worth families, it is extremely common for an institutional trustee to be named. Most major banks and investment houses, and many smaller ones, have trust departments that do nothing but serve as trustee for large trusts. Anyone of mental competence having reached the age of majority, may serve as a trustee with full powers as granted by the trust instrument.

Yes. The other attorneys responding here are all correct. Sometime a settlor does not wish to serve as his/her own trustee. In my experience, the most common reason is that the settlor may not wish to continue to act as his/her own trustee is that he/she is elderly or infirmed. As long as the settlor has the requisite mental capacity and is not being unduly influenced, the settlor may choose any individual or individuals to act as trustee or co-trustees in place of the settlor him/herself.