Just help her pay for the dog… woof

The child is still a child and you might remind mother that she is under court order to insist that the child be available for visitation. I...

If you mean present at the hearing following the petition for name change, no; filing a notice of no objection would be helpful though.

If she’s been personally served, all subsequent documents only need to be mailed by anyone over 18.

You need to file a Responsive Declaration (FL-320): https://www.courts.ca.gov/documents/fl320.pdf

He can continue to complete the process; here’s a link to instructions:

She doesn’t include imputed income - only any changes to actual income or expenses.

You can use a caregiver affidavit which can be revoked in writing; it does not involve the courts: https://www.courts.ca.gov/documents/caregiver.pdf

Generally, that would be the expectation of ‘making a deal’ but the terms of the deal can be anything to which that both of you agree.

You would call it a Reply and put it on pleading paper.