If they allow transfer of visa to new passport there should be no fee. If British, for example, there is no paperwork with a new passport from VFS who handle passports these days.
Have a search through group posts. There has been someone posted in the same situation ie stolen passport rather than expired where the holder has both old and new passports. Find out the exact process you need to follow.
Sounds exactly like the visa for you. I would mention you would be doing some work remotely while in Thailand - even if just providing quotes and invoices. A pal did this for his UK Building company
As you say there should be no issue. I have done similar on different visas. I was in and out on Visa Exempt 4 to 6 times a year when working in Belgium. Up and down 8 to 10 times a year when working in Singapore. When working in Thailand travelled about every 6 weeks. They have periodic tightening up but for genuine travellers, visitors and visa holders there are no issues.