Depending on your architecture, there may be different mechanism for the u-boot to communicate with the Linux kernel.
Actually there may be some structures defined by u-boot which are transferred to and used by the kernel using ATAGS. The address in which these structure are passed is stored in r2 register on ARM. They convey information such as available RAM size and location, kernel command line, ...
Note that on some architectures (like ARM again) we have support for device-tree which intends for defining the hardware in which the kernel is going to be run as well as kernel command line, memory and other thins. Such description is usually created during kernel compile time, loaded into the memory by the u-boot and in case of ARM architecture, its address is transferred through r2 register.
The interesting thing about this (regarding your question) is that u-boot can change this device-tree structure before passing it to the kernel through device tree overlay mechanism. So this is a (relatively) new way of u-boot/kernel communication. Note that device-tree is not supported on some architectures.
And at the end, yes, the hardware is reinitialized by the kernel even in they have already initialized by the u-boot except for memory controller and some other very low level initialization, AFAIK.