In your results you can see that where you have tied the surfaces together, heat is being conducted away but only at discrete positions where the nodes on the very coarse mesh connects to the refined mesh of the smaller block. In that case you'd need to refine the larger block.
Where you have apparently made a single part it appears that you can see that the temperatures aren't equal at the intersection of the 'two' parts, for some reason. The contours of NT11 should run across the 'two' parts but don't. It suggests that the single instance is a still two parts.
The instability in the solution is common in severe transients and does damp out as the solution continues through time. Try modelling it in 2D or 1D and put a very fine mesh, biased twards the point where the heat flux is applied and you should see an improvrmrnt. My guess is you'll still see some instability where, mathematically, temperatures appear to reduce below the position in which the flux is applied. I tend to ignore these discrepancies where they occur in the very initial period of the transient as they quickly disappear and the solution smoothes to a more reasonable result.
Tata