Are there any Linux tools (command line, or Live CD) that will test hardware components for errors (hard drive, RAM, CPU, graphics card, buses) and show me some detailed output with wrong parts etc.?
There are many tools that specialise in testing different components. No all-in-one tool does currently come to my mind. (Also, if it exists and does not internally use any of the well-established tools, that would be stepping against the principle of modularity, which is one of the fundamentals of Unix philosophy.
For RAM testing, I recommend Memtest86+. You have to boot into it rather than your primary system (pretty obvious if you consider the role of memory to a running OS).
For hard drive testing, you can try:
I don't recall any generic tools for testing components such as CPU or a graphics card.
As a general tip, I recommend using a specialized live distribution such as SystemRescueCd, that is released specifically as a tool for resolving major system problems.
For memory, there is Memtest86, though it is not strictly to be considered a Linux tool:
Memtest86 is designed to run as a stand-alone, self-contained program from a bootable floppy disk, CD-ROM, USB Drive, or from a suitable bootloader without an operating system present.
Popular Linux distributions such as Knoppix and Ubuntu offer an option when the CD starts to run Memtest immediately (instead of booting Linux)