Hi imVert,
Before you go in the direction of designing and building your own DIY mattress I would read post #15 here along with post #2 here.
Egg crate foam that deep in a mattress may have some small effect but not a lot because most of the warm air wouldn’t penetrate the 5" of foam above it. You can read more about the factors that can affect the sleeping temperature of a mattress in post #2 here.
There would be no problem with the latex over the eggcrate in terms of durability.
I would also keep in mind that the mattress you are designing would be very different from the Tempurpedic in terms of how it “feels” and performs. Slow response foams (like memory foam or gel memory foam) and fast response foams (like polyfoam or latex) are very different from each other and closer to being opposites than the same. There is more about how they compare in post #2 here.
Phoenix