Well I can answer the second part of your question - yes, the heat from the wall is a factor, along with rainshadow, meaning the bit directly below the wall may not be receiving any rain at all when it rains, or much less than the rest of the area.
As for the 'problem' grass, what I can see in the top picture is a few sparse blades sticking out of what appears to be sand, and sand with a hole in it where either something's been inserted, or an insect is entering and exiting. If the second photo is supposed to be showing a different type of grass, I'm afraid I actually can't see any difference other than a thicker covering, so a bit of clarification and possibly more pictures showing the grass you want to keep and a bigger patch of the grass you don't. Usually, though, when a weed grass invades a lawn, there is no easy treatment plan - any weedkillers designed for use on lawns are meant to kill broadleaved invaders, not rogue grasses, so the answer has to be the old fashioned one - dig out the areas of rogue grass and replace either by seeding or turfing the patches.
