Is this processor compatible with all motherboards that have an LGA1200 socket, or are there specific models recommended?

The processor is compatible with Intel 400 series chipset based motherboards.

Does the Intel Core i7-10700KF require a separate GPU since it doesn't have integrated graphics?

Yes, the Intel Core i7-10700KF requires a separate GPU as it does not have integrated graphics.

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Intel Core i7-10700KF Desktop Processor 8 Cores up to 5.1 GHz Unlocked Without Processor Graphics LGA1200 (Intel 400 Series chipset) 125W