... Thai culture training is normally included in TEFL training done in Thailand, but no, not required for renewing a work permit, unless this too has now become a new requirement.
... To my knowledge, your new landlord need to register you as the new tenant at Immigration and when he/she did that, you won't have any problems. If you visit immigration for 90-day reporting, and your landlord did not register you as his/her new tenant, then you have to wait until they do ... and a penalty is due to the landlord. It happened to us moving from Bangkok to Pattaya (Chonburi) and at the end I paid her penalty of 1600 Baht because she lives in another province.