Is the short form of "sun" 해 not acceptable here?
I think their idea is that the sun itself is always hot but the sunrays or sunlight isn't, so you would have to specify like they do here.
Duo is just sloppy. "Sun" is not the same as "sunlight". The clue here is that they put the subject particle on 햇빛이, whereas they didn't on 해, so in their eyes just 해 is not grammatical.
I don't quite get what you mean, but we don't say "the sunlight is hot today" we say "the sun is hot today", if you translate it literally it wouldn't make sense either, would it?
What is 햇빛이?
(From naver dictionary)
①sunlight; sunbeam; sunshine ②seeing the light
(이/가 are subject particles)
햇빛, in here 해 means sun and 빛 means light, ray.
Isn't 뜨겁다 more used for things that are hot to the touch?