I presume you are referring to a non-o visa based on retirement. There is no requirement for an onward ticket as that only applies to visa-exempt and visa on arrival. You will be required to prove you are in Vietnam so use your entry stamp and accommodation booking for the address. You will need pdf copies of your bank statements (they do not accept screen shots)
It most certainly is a rule for visa-exempt entry, but they only ask if they suspect you are not a genuine tourist and select you for secondary screening
You get stamped in for 60 days every time you enter during the 6 month period. If you arrived the day the visa expires you still get 60 days and each 60 day entry can be extended by 30 days at immigration. You can leave and come back as many times as you like. It is preferable to fly out and in but it can be done at a land border. You can exit and return the same day so no need to spend 7 days in Singapore unless you really want to. If you time the exit and entries carefully he can legally stay almost 9 months with an METV. If you plan to extend 30 days then go in to immigration and ask what is required.
You can get a non-O visa based on visiting family, but you can not extend that for 12 months unless you have guardianship of your daughter via the family court. So getting a bank account is moot.