Is the Intel Pentium Gold G5600F compatible with motherboards that support the LGA 1151 socket from previous generations, or is it limited to the 300 series?

The Intel Pentium Gold G5600F is only compatible with Intel 300 Series Motherboards.

Does the Intel Pentium Gold G5600F require a dedicated graphics card since it comes without integrated graphics?

Yes, the Intel Pentium Gold G5600F requires a discrete GPU because it has no integrated graphics.

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Intel Pentium Gold G5600F Coffee Lake Dual-Core, 4-Thread, 3.9 GHz LGA 1151 (300 Series) 54W BX80684G5600F Desktop Processor Without Graphics