Hello Caragon
This is quite a common issue. Actually a few things you should know.
Yes firstly the China airlines staff are quite correct in that super cheap tickets don't allow seat selection in Economy.
Also I noticed you said you will be flying in November. China Airlines does not do seat selection more than 90 days ahead. So while the staff member was correct she/he did not mention that to you as well. So even if you bought a business class ticket you would not be able to seat select until early September at the earliest.
There are 2 options I see here.
Wait until check-in and get there early, it would be good to also ring the office 3-4 days before hand and admit your ticket it is not possible to seat select beforehand but to ask can let their agents to know this issue. If checking in early agents will have the ability to move around if needed especially single travelers wont be too much trouble. Even checking in late they will make sure 2 of you are sitting together with your youngest member being accompanied.
Anyway that's why I suggest going to arrive 30 mins before check in opens. You will be surprised many people don't bother doing seat selections, plus November is not a particularly busy time so while not impossible the flight is almost certain not to be completely full.
If your still concerned greatly and cant sleep then Sky Couch is the only way to go. It will guarantee all are together as it is done with 3 seats together which can convert into a short bed. Good length for children but too short for adults to really lay down, though you can put your back to the window and stretch out that way. Hence the word couch.
Note that Sky Couch will only be on your USA- Taiwan flight. Taiwan to Thailand is not stable though I do believe the early morning flight is the B747 most of the time. Also its only a 3hr flight which is short haul.
I am sure it will work out fine.
Hope you know that B777 (US-Taiwan) configs are quite tight. New plane but still.