What is the purpose of this interesting pattern of class E airspace in northwest Alaska?
This is easier to explain in reverse: rather than why there is so much class G airspace there, why is there so much class E airspace  everywhere else?

Remember, all airspace is class G unless it has been designated something higher. Wind your clock back to the early days of aviation, and the entire country was class G!

However, that isn't very safe, particularly in bad weather. Pilots were getting lost and running into each other. 

So we got navaids and radios, and narrow strips of airspace along between navaids became controlled airways, i.e. class E. If you scroll over to Canada in SkyVector, you can see what this probably looked like.

Fast forward several decades, and there are now so many airways criss-crossing the US that they've merged into one big class E blanket, so it's easy to forget how we got here.

But there are still a few areas where the airway density never quite reached full saturation, mostly in mountainous terrain (of which Alaska has a lot), and those pockets still remain class G--and reveal the past.

As to why the V airway in your example has a different floor than the T airways, I'd assume that comes down to the minimum VOR reception altitude due to the terrain. Other than the floor, though, it's the same.