Two options. AIS or True. Not much in them if you’re in the main tourist areas. Perhaps AIS have a slightly better rural coverage but it’s going to be pretty close.
There’s no problem with you travelling with her per se, but because you don’t share the same last name then you’re probably best to travel with a letter from your husband and a copy of his passport that says he allows you to travel with your daughter. Perhaps you’ll never be asked for it but it’s best to have in case you are.
Yes. You only report if you’ve been in Thailand for 90 consecutive days. If you leave prior then the need to report ceases and restarts when you return.
If it’s your first extension from a Non O then they’ll usually put a small piece of paper in your passport notifying you of your next due date like this.
If it’s not your first extension then your 90 day requirement doesn’t change from the previous one. That’s due 90 days from your last report regardless that you’ve just got a new extension. Or 90 days from entry if you’ve been out of the country