As an added bonus if anyone can explain why to use Мне холодно instead of Я холодный I will be ever so grateful :P
Why do you say in English I'm cold, while you are the same temperature?
In Russian you will become cold soon after the death, when your body becomes the same temperature as the environment.
Coldness can be felt only by an external observer, when he somehow feels that the object has a lower temperature than him or than he expects.
(You can sometimes use I'm cold, when you talk about you character).
When you say I'm cold, you are not cold but the environment is too cold for you, which is expressed in Russian. Empty it is not used in Russian because Russian allows subjectless sentences. The verb to be is omitted in the Present tense. For or to me is expressed by the dative.
Whether it is an adverb or the short form of an adjective is irrelevant here.