That's a multi entry tourist visa METV. It last for 6 months but if you time your border bounces you can get nearly 9 months. On your last entry there is nothing to stop you leaving and returning then getting a visa-exempt entry. They don't do a one year visa now.
You'll both need re-entry permits to keep your permission of stay valid. The baby will get a free 60-day visa-exempt entry. Children under 15 do not get fined for overstays should you have any problems in the future.
Just book in a cheap hotel for one night before going to do your extension. You only have to take a printed copy of TM30 if you stay in private accommodation. The hotel will do your TM30 online so just take proof of your booking.
Tourists can work online if their work doesn't involve any business in Thailand. As a UK citizen, you will be stamped in for a free 60-day visa-exempt stay.
Christmas is not an official holiday in Thailand, but they close for New Year. If you are staying in a hotel it is easy but if you're staying in private accommodation you will need a TM30
If you adopted her children you can stay on a visa based on Thai children but that stops at 20 years old. So as her children are in their 30s you will have to convert to a retirement visa or leave Thailand.
I've seen this question raise over the years on numerous forums and it has been established that minor maintenance work in and around your family home does not constitute work.