Q: Do I have to tell my child their apple ID password to use Ask to Buy? I'd rather they not be able to change settings on their phone such as Find my Phone.

simple solution

goto setting/general/restrictions/

create a passcode that you will not forget and they will not guess (this passcode continue from backup to backup and is impossible to bypass)
go the near the bottom

choose accounts

check don't allow changes.

you can also set a lot of other restrictions here, especially if your child has been mis-behaving.

WWhat Daniel said here the knowledge base article About Restrictions (parental controls) on iPhone, iPad, and iPod touch - Apple Support