1) If you live with the child in California, then that is where you need to file for child custody.
2) Although I seriously doubt this, the mother herself can ask the judge to order a child support
order for her to pay. So even if you do not wish for her to pay child support, the court does have the discretion to order support.
3) Only way to cut off a parent completely is for the parent to have her parental rights
terminated. She can voluntarily agree to this or you will need an attorney to petition the court to terminate her parental rights. In general parental rights are not terminated unless there is a stepparent adoption
process underway or there is evidence of clear disinterest, failure to maintain contact with the child. In general few attempts to contact the child maybe enough for the judge to state that she has made attempts at having contact. This is why if you decide to terminate her parental rights you would need a lawyer to represent you.
4) As to custody, in California it is presumed that joint custody
is best for the child unless it is clear that the other parent is a detriment to the child's best interests. What is a detriment depends on whether the mother is unfit to parent the child. Based on the facts you have provided, since you have been the primary caretaker you would be able to get custody of your child, the mother can be entitled to visitation unless you show that she is a risk in taking off with the child, if that is the case, she can be ordered supervised visitation
with you or a relative. But if she has had few visits with the child, the court can order visitation unless you are successful in terminating her rights.
Below are the statutes for California on how custody is addressed for your review:
PART 2. RIGHT TO CUSTODY OF MINOR CHILD
CHAPTER 1. GENERAL PROVISIONS ................................. 3020-3032
CHAPTER 2. MATTERS TO BE CONSIDERED IN GRANTING CUSTODY ....... 3040-3048
CHAPTER 3. TEMPORARY CUSTODY ORDER DURING PENDENCY OF
PROCEEDING ......................................... 3060-3064
CHAPTER 4. JOINT CUSTODY ...................................... 3080-3089
CHAPTER 5. VISITATION RIGHTS .................................. 3100-3105
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