December or January won't make any difference. But I would suggest you look at getting an appropriate visa. Visa exempt entries are primarily for the purposes of short term tourism.
you can't be self employed and working inside Thailand. If you mean you will be working remotely from Thailand then the DTV is really your only option but you can't work for any Thai entities or clients that are in Thailand. At your ages your options are limited visa wise for a stay of 1 year, even if you are not working the DTV is still probably your best option by signing up to one of the soft power activities with an approved provider (Muay Thai, Thai cooking or for medical treatment)
Assuming you are on extension of stay based on working you should just be able to change the reason for your extension but will need to cancel the work permit
Bebbieluu depending on what assessable income you bring into Thailand you will need to file and pay any Thai tax due. If the country where you otherwise pay tax has a dual tax agreement with Thailand you can claim a credit for any tax already paid on that income against Thai tax as part of the process
Nobody can tell you that without a lot more information. Nationality, ages of all, purpose of visit and so on. Start by researching the appropriate visa options on the Thai evisa website