I have heard that fire magic is having some issuse in the new area, but is Frost or Lighting having the same issues?

Become a storm mage. Chain Lightning is probably the best spell there is. Thunderbolt isn't half as good. Great against mages.

Lots of Dunmer - 50% fire resist. I assume ash spawn have fire resist. It is almost as if they decided to make life difficult for people who had specialized in fire. I would go with lightning while on Solstheim. Where I normally enchant my weapons with fire, this time I went with shock because it is more useful both in Skyrim and on Solstheim.

How good is Frost in Solstheim? Heading there with Frost-mage vampire, though I have unused perks in case I should boost damage in shock spells, which seem most favorable

I use my frost and shock magic mostly on Solsheim.

I would go with shock for a couple of reasons. Especially if you are choosing just one school.
- It has the fewest creatures resistant to it when compared to the other elements.
- The disintegration perk is better than the other two elemental perks.
- Shock spells travel speed is instant so it is much easier to hit moving targets.
- The strongest elemental atronochs you can summon are Storm Atronochs so you can summon them and fire chain lightning into the crowd without worrying about damaging your atronochs.
- Lightning storm is the most usable of the master destruction spells and is great for killing dragons.