1) The simple answer is that romanized hangul is inaccurate and shouldn't be used to learn any aspect of Korean. Use hangul and your ears; don't use romanization. 

2) ㅋ is pronounced like k in english, ㄲ is pronounced like a tense/fortis g in english, and ㄱ can sound like either a k or g depending on its placement.

It is very arbitrary how you should transliterate Hangul to Roman letters. The reason why I would write 'g'  for  ㄱ is because it sounds like the 'g' in 'to go' in English. It is softer than 'key'.  But almost no sound in Korean (or any other language) can be mapped 100% to a sound in English (or any other language).  

Romanization is just 'personal' I think, look how I map the following lyrics for my own use :)

숨 소리가 들려, 몸이 녹아 내려

<font color="#000000">Sum  só.ri.ga  deur.lyò,  móm.'i  nóg.'a  nè.ryò</font> 

In writing Hangul (Korean characters) there is a correct spelling I think, but not really for romanization.