yes, it can run lower resolutions at higher refresh rate, as long resolution x refresh rate has enough bandwith
4k at 60Hz is 18Gb
1080p at 144hz is 9Gb
1080p at 240hz is 18Gb
there might be pixel clock limit, which can limit refresh rate...but that can be usualy resolved with firmware update, but since it says 4k at 60 works, then 600MHz pixel clock is enough for 1080p @ 240Hz
First off, just to be clear, the motherboard alone doesn't supply any video output, you DO need a CPU with iGPU to get any output at all.
All the resolutions and refresh rates are listed in the hdmi specs so you would have to look up what version of hdmi the mobo has and you would also have to see if the CPU supports that output as well.