Medical doesn't seem to be a word people use in their posts. Instead search for something related like "doctor", "dentist" or "hospital" and you find many posts.
They don't take your current passport. They cut the corners to indicate its invalid but you keep it. Then you go with both passports to immigration to move visa. I am belgian too and I have done this in thailand.
I hear you, bit similar for me but location proof is a required question in the online application but whether they really check is up to the embassy. I guess we got lucky. But you know, the other day there was a post here of someone that had to provide a picture of himself with a recent receipt in the country (I think it was Vietnam) so it really does happen but possibly only if they suspect something is wrong.
There is also the matter of possible interview so imagine if you are invited and are not in that country..
Because each embassy covers a certain area that they are responsible for. And you can only apply when you are in their area. For example if you are in Laos then you apply through the embassy in Laos. If you are in your home country then you apply there. You have to proof your location during the application too so you really need to physically be there. Good luck.
I wrote an employment contract for myself which had a clause that says I can work remotely from anywhere with an Internet connection. The contract was signed twice by myself. Once as employer and once as the employee. Accepted without questions.