If you cancel your permitted to stay inside Thailand on day xx/xx/xx. you have to leave on that day xx/xx/xx and not 6 days later or whenever you choice.
So in your question cancel on Jan 3 means leaving Jan 3 before midnight, as the organization is willing to adapt the last day of you working for them, pich a day so you have time to pack your bags and leave after you also cancelled your work permit.
You need a stamp like this, it can be a single entry or multiple entry, check your passport or buy one at local Immigration OR just before departure on airport
YES, you can apply for the 1 year Extension of Stay based on employment in Thailand.
BUT you need to meet some simple requirements, like salary payments to staff, paying income tax and a few more, visit your Immigration Office to get the list of requirements, tell your staff to start up the copy machine, tick the list when copy is made and when all done, go back to apply.