A simple explanation is that a "visa" is used to enter a country, that's all. Once in a country you get stamped in until a specified date, which can usually be extended. The visa is long gone.
I did because my son will most likely be living in Thailand and going to school here for a long time. I didn't want his passport full of overstay stamps and it was easy enough to just include him as my dependent
not in Bangkok. I submitted my T28 about 6 months after I actually moved (although I put a more recent date on the form), sent it off with my regular 90 day report and got both receipts back without a problem.
You'll be fine with a ME Tourist visa. Your kids should have visas too, although if they don't they'll still be stamped in for 30 days, visa exempt, and then stamped as overstaying when they leave. But there is no fine or ban for kids who overstay.