Your application I believe goes to the Thai embassy in Rome for processing. They recognise Cyprus as being the entire Island. So you should be able to enter your Turkish passport and your location as Cyprus
If you are genuinely receiving medical treatment you will be able to supply a treatment plan on the official paper from a hospital. Your use of clinic implies it's not a hospital so you will probably struggle
If you have an employer they want a contract that shows you have the right to work remotely from Thailand. If you have your own company the registration details. Otherwise you could show them a tax return that shows you pay tax in your home country.