This visa allows multiple entries and with each entry you will be allowed to stay up to 60 days (extendable by 30 days for 1900 Baht in country). You can even enter on the last day and will still get a stamp for 60 days (also extendable).
You want to get a new METV or Single Entry Tourist Visa (SETV)? METV can usually only be obtained in your country of residence.
It depends on the embassy/consulate. In Laos and Malaysia for example, it’s ready the next day. In Cambodia it takes about 3 days. So where do you plan to go?
According to some online posts the mother of the children should go to the amphur with the birth certificate and get a letter from there, that states, that she gives her consent that you can travel with her children. Maybe she can call the amphur and ask there?
Maybe also get that translated for the immigration in HKG?