in study 2, you have no way for the heat to get out, so i could see the solver being upset with trying to solve your problem. you need to apply some kind of BC like heat transfer coefficient on all the faces. Otherwise it is assumed to be completely insulated on every face with heat generation inside and 21degrees starting. Your answer is infinity!
on the first one, what element size are you using? if it isn't 1/2 of the thickness of the part, you're not going to get good results anyways. curvature mesh is a good idea to try and get it meshed and you might get lucky with a more coarse mesh. but your analysis has some flaws before you even get to meshing and running. your parts aren' touching. so the heat from your cylinder won't make it to the heat sink. and then i'm not sure what your plan is on the face boundary conditions. this needs some though.
i'd suggest starting with a block with some fins to get comfortable with setting up the analysis and running it. you should also be able to hand clac the results so you're confident in them. then you can go to your model.
consider that you should be able to use either half or cyclic symmetry on this problem depending on your assumptions. this will reduce your meshing headaches significantly and get you quick results.