I checked 'vemodig' and my dictionary gave 'längtande' as its synonym. That sounds like 'yearning', a sense of 'nostalgia' (and a perfect translation for Finnish 'kaihoisa'), but I have no idea if 'vemodig' means the same.
Yes, feeling blue is a very similar feeling to vemod, but there is often a nuance of regret, longing or nostalgia in it too. Literally, vemod means woe mood. The sort of mood when you'd say woe in old English.
The closest I can think of is melancholy. A sad yearning or longing, usually for an unattainable element or experience in the past. It is more nuanced than that so it is flexible to fit a range of moods.
Yes. The "G" is pronounced more like an English "Y" ie almost like sorry-lit... that is a very coarse description though and not the right pronunciation, more a demonstration of the structure of the word.