When I buy a barebone with "Integrated ATI Radeon HD3000", do these onboard graphics work with every CPU?
I am asking because I want to build a backup server, and I want to use an older, not so expensive CPU for that.
No it does not work with every CPU. First, it will only work with processors made to fit the socket on the board. Second, only processors that are supported by that version BIOS will work with that board. This does mean that with some boards, a processor might be the correct type to fit into that socket, but might not be supported by the BIOS (or even board design).
An example... Many manufacturers make socket AM2 boards that will support 95 watt AM2 processors. However, the more powerful 125 watt AM2 processors will not work with those boards, even though they are made for the socket on that board. More specific? This Asus board has a socket that will fit the AMD Athlon 64 x2 6400+ processor, but that processor is not supported by the board.
So... you need to go to the motherboard manufacturers site and examine the product page, and view the support CPU list for that board, to know what processors are supported for that board.