You ask a few different questions here. First, a “note” is not a legal court order and will not change the obligations imposes by the order. While it may protect dad from accusations that he is violation of the order, it would not entitle dad to keep custody if mom decides to resume it. The better practice is to ALWAYS get a court order to conform to the new agreement. This is not particularly difficult when the parents are in agreement and you can proceed by stipulation.
As to child support, it cannot be waived. The right to support does not belong to the custodial parent, it belongs to the child. So even an expertly written notarized contract signed in blood will not waive a future request for support. An attorney can help you to set this up so as to avoid future turmoil when mom changes her mind- again.
Best of luck to you.
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This response is intended to be a general statement of law, should not be relied upon as legal advice, does not create an attorney/client relationship and does not create a right to continuing email exchanges.
As for your custody question, it's not clear from your comments whether Mother has already written a note stating that she wants Father to have sole physical custody. I'm thinking not, but instead, that she only made overtures via telephone. If only suggestions via phone, one has to question how serious Mother is about her proposed custody change.
Assuming Mother is firm in her resolve that she wants Father to have sole physical custody because he can provide a better environment for their daughter (which is not typical behavior, although it does happen when parents truly act in their child's best interests), and thus will not change her mind, then the best approach is to hire a local family lawyer to draft a Stipulation and Order Regarding Custody Modification. Such Stipulation should contain terms modifying physical custody per the parties' desires (i.e., along the lines you stated in your question). Mother could then sign the Stip & Order, which would prevent any problems if she later changed her mind about the custody change.
Although parents can change custody without a Stip and Order, it's not advisable. As the other attorney said in her answer, if Mother later changes her mind and demands a return to the prior custody arrangement, a nasty dispute could ensue creating problems for all involved, including the 14 yr old daughter. E.g., it could create a problem with switching school, including uprooting from from friends and social networks, after-school functions, like band or sports teams, etc. It's not good for children to be repeatedly uprooted. 14 is already a tough age for teenage girls. No need to bounce her from one school to another, and then back again, or inviting a custody battle. This custody change should not be made on a whim.
If you're worried about Mother changing her mind before signing a Stipulation and Order (i.e., the few weeks it may take to hire a lawyer and draft the Stip & Order), consider that you will likely have avoided a problem that would have occurred down the road after the informal custody change if/when Mother changed her mind. I.e, if Mother changes her mind in the short time while you got a Stip drafted, then she was not firm about changing custody, leading to the problems described above.
As for your child support question, parties cannot agree to eliminate a child's right to support. However, if Father earns a high income or has a level of wealth in which the child's interests will be adequately taken care of while in his sole physical custody, without any child support paid by Mother, then the parents can agree to a "non-guideline" support order. I.e., they can agree that Mother will not pay any or very little support until further agreement or court order... again, assuming such agreement will not cause any detriment to the child because Father can adequately pay the costs related to raising her.
Remember, child support can be modified at anytime up until the child is 18 (or 19 if still in high school) if/when there are a "change in circumstances." Cut and paste the following link: http://www.courtinfo.ca.gov/forms/documents/fl192.pdf . This is Judicial Counsel Form FL-192. On pg 2 it provides instructions on changing child support.
Finally, as for back support (child support arrears), the answer is "it depends" the circumstances. I.e., is Department of Child Support Services involved? Has the child ever received county or state benefits, such as Medical, with an amount still due for such services? If not, is Mother willing to agree to an amount that is less than the arrears and accrued interest?
You should discuss this particular issue with a local, experienced child support lawyer, especially if there are significant arrears and accrued interest. Remember, child support is the child's right, and parents can't eliminate that right. But, if the child's expenses are adequately covered, exceptions can be made to child support arrears via a parents' agreement.