Is being a pure Mage viable in Dragonborn land?

Definitely.

I'm doing it right now with a Dunmer Mage. He's in my sig if you care to let your gaze drift down a bit.

I'd recommend hiring a follower, and to bring Cold magick, but it's definitely viable.

Of course. However, keep in mind that your time in Solstheim may be a bit difficult if your mage is a low level character.

I would also recommend bringing Frea or Teldryn along just in case things get a bit claustrophobic for you.

It depends a little on your setup but yes, most viable. Basically if your mage can handle most challenges in Skyrim they will be able to handle Solstheim as well; if they're already struggling on the mainland though they're in for a reasonable challenge. Fire magic specialists are at a bit of a disadvantage as quite a few enemies resist it, but I've personally done it and only had to switch to a different type of spell on only two occasions.

Destructionists get a badly needed new way to amp damage as well (three new dragon priest masks that are elementally aligned), and the final All-Maker Stone is a great boon to those who don't want to commit crafting abuse to make spells affordable to cast.

Also this.

For most characters, I recommend not going to Solstheim until you're at level 20 or so. For a pure Mage, since the enemies are tougher and with the way all magick in the game scales... I'd say closer to level 25 would be a good bet. Those few extra Perks tend to make the difference without breaking a build just to survive.

I'm starting over as a Wood Elf, Mage.
Or at least I think I am a Mage.

My character will focus on Illusion, Restoration and Destruction.
I thought about not dwelming into too many schools as I would probably level too fast then.

I don't want to be using Conjuration as I prefer interactions with mercenaries, real humans.

I hope you have good Conjuration and Alteration skills...

Good choices, but I'd also toss in some Conjuration. Those extra "insta-reinforcements" can really save your tail at times. If nothing else, it helps to even the odds when you get jumped by a gang of enemies.

And that will happen.