Does Apple design its own CPUs?
Apple is known to design the ARM used in the iDevices. But Apple definitely does not design everything. For example the GPU is a PowerVR G6430 and the ARM architecture is derived of ARM's V8-A.

On the Desktop/Laptop it's done by Intel since it's their CPU. 

Apple's latest A7 are made by Samsung, but it's believed to shift to other manufacturers (for example TSMC) since Samsung is a major competitor for Apple.