They dont Transfer any Visa (you keep both Passport) - If you on extention stamp that should be Transferered but If you have to get new stamp soon just wait
As there ist No 30 days Visa he MUST be on Visa extempt - he can extend / Not important where He stamped in - he must Go to Immigration where He is registered for staying
She need to - but If she dont - better you do IT - at the end the Problem is with you when you need Extension (our contract says WE have tondo ourselve but First time Landlord Register us - But now when WE come Back from abroad i never ASK him - Just do myselve so i can BE Sure ITS done within 24 hours
and Covid Extension is unable to travel / so with your comment “need for future work” it is clear you are not unable to travel and extension would most likely not granted - so very helpful from the officer asking you to remove that
Well in fact your research was wrong and you tried to talk to the immigration as you know better than them - no wonder they where not “kind” - but they did everything right - the 60 days can be granted after the first 30 days EXTENSION has been granted