What's the meaning of 'UOS'?

The 'UOS' means 'micro operating system'.

Which LPC800 is the best platform?

The best platform

The best platform is LPC824 (ROM:32KB, RAM:8KB).

Minimal platform

You can also create an application with LPC810 (ROM:4KB, RAM:1KB).
However, you should change the stack size for the tasks and should keep only two or three tasks in your system.

Is it possible to use the kernel on ARM Cortex-M3 or M4?

Basically, The answer is yes because the kernel operation codes are using M0 operands only.
M3 and M4 operands are covering M0 operand you know.

How do I change the implementation for my own board?

You should create a new implementation of the board_[YourBoardName].c such as board_lpc800_miniboard.c and board_lpcxpresso_lpc812.c.

How do I use the kernel on LPCOpen Software Development Platform?

The software development platform using CMSIS in it. However, you should change the include header from #include "LPC8xx.h" to #include "chip.h" in uos.c.