Vinyl Plank Flooring - Questions

I'm replacing the old cork floor in the ground level basement and I think I want to go with vinyl plank flooring. The new flooring has a underlay built in, and the subfloor is concrete. I removed the old baseboards and quarter round along with the cork floor.

Two questions:

1) The cork floor was super cold on my feet in the winter, but I need this space to be my primary office. Given that, will the vinyl with underlay be warmer than the old cork flooring? If not, will a thermal underlay help out or will it cause me more issues down the road? Most of my research seems to point to not needing any other subflooring on the concrete as the vinyl I want has it built in and it might actually void the warranty, but I want to be sure.

2) The cork flooring is twice as thick as the new vinyl I want to go with. After installing it there will be a 1/2 inch gap between the vinyl and the bottom of the drywall. I don't think this would cause any problems as I can cover it up when I put down the new baseboards, is that right?

I've added a picture of the previous cork floor and new vinyl I want to go with, illustrating the gap: https://imgur.com/a/uRLqSVr

 

Cork has a good R value but it's still down in the R2-4 range so it's not going to provide a lot of insulation value!

However, to gain the most of an insulated space you go up, out, down, meaning the floor which has the benefit of the earth constant heat has the least impact.

Want to insulate the basement, look at the walls!