Can unregistered partners and their child apply for a DTV Visa to Thailand together?

As you are not married you would both need to qualify for the DTV independently. If you are granted a DTV you could apply for your kid to get their own DTV as your dependant

First of all, there is no such thing as being on the same visa or adding someone else to your visa. Each person has to have their own visa. The way the DTV works is that one person can be the primary visa holder, and they must apply for and pay for their DTV. Then once they get it, a dependent DTV can be applied for based on this initial DTV, and one of the requirements is to upload the primary DTV as part of the documents. Each one must be applied for and paid for separately, and the dependent DTV is only available to spouse (legally married) and children.

Whether your daughter requires an education visa or not is up to the school. Some will want to know the child is either not on overstay, or for them to be on an ED visa, and some won't care either way. Most people with children that age do not do anything with visas and just let them go on overstay as children under 14 cannot be fined for overstay and children under 18 cannot be banned for overstay.