I would agree that "cute" is a bit subjective, but the "cute" fonts I've seen are generally somewhere between a "proper/formal" font, and the handwriting of a child, where you find the imbalances in spacing and size, which are common to children or people just learning to write.
In addition, one could associate thick, rounded strokes with rounded ends and lack of serifs in that colored font:
versus one that doesn't have those qualities:
The fonts just replicate certain aspects of that imbalance while still maintaining legibility.
Also, is it acceptable to write a 'ㅎ' with just one horizontal line above the 'ㅇ'?
No, that would be like writing a K without the one of the lines. It wouldn't be considered a letter. :)