Is this CPU compatible with motherboards that support LGA 1151 sockets, or are there specific chipset requirements?

This CPU is only compatible with Intel 300 Series Motherboards.

Does the Intel Core i5-9400F processor come with integrated graphics, or is a separate graphics card required?

A separate graphics card is required as the Intel Core i5-9400F processor does not come with integrated graphics.

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Intel Core i5-9400F Desktop Processor i5 9th Gen - Coffee Lake 6-Core 2.9 GHz LGA 1151 (300 Series) 65W Intel HD Graphics 630-BX80684I59400F

9th Gen Intel Processor
Discrete GPU Required - No Integrated Graphics
Only Compatible with Intel 300 Series Motherboard
Socket LGA 1151 (300 Series)
Max Turbo Frequency 4.1 GHz
DDR4 Support
Intel Optane Memory and SSD Supported
Intel Turbo Boost Technology 2.0 offers powerful performance for mainstream gaming and creating