When you turn the nova into a channelled ability it also gets the “channeled” tag in the description.
How this works with dragon mage, I don’t know. perhaps you could figure it out by yourself, casting while looking at the stats screen.
Fire, ice and cold are all elemental damage. So despite it not being on the skill tag, elemental damages scales with these damage types.
Also what you write about damage over time is correct. It’s similar to crit. We have more than 1 source to increase damage on this. The more specific a damage modifier is, the higher it’s possible values. For example elemental damage over time > damage over time, because the latter is global while the first is specifically only for elemental damage.
I last tested that in 0.7.7 but yes, it worked with the damage buffs from the Dragon passives, but from memory they are a single buff (per element) if you have ast the relevant element in the last 4 seconds, they don’t stack.