I can't tell if it's supposed to be like a plate of metal with lines drawn on it, or just the grid shown. I guess you've tried to solve for the grid shown, but I think you've got it wrong. There is a symmetry, which shows that T2 and T3 should be 150 (for either problem), in which case, T1 would be 125 and T4 would be 175. I think your initial equations are right, and I don't think your solution satisfies them.
In the case that it's really a 2d plate with boundary conditions along the edges, you are actually solving Laplace's equation, which is the time-invarianct version of the heat equation. It's not hard to solve numerically using a fine grid and solving a big linear equation. Analytically, the solution will be the real part of some complex analytic function, which I don't recognise. Near the TR and BL corner, the solution will approximately depend linearly on the angle from the corner.