In practice: those which come with a HDMI connector onboard (or perhaps DVI + external DVI->HDMI adapter?).

Unless your hand soldering is good enough to obtain correct impedance & match wire lengths.

I'd expect most such boards to be capable of generating a 4K signal timing-wise. But have enough LUTs, blockRAM etc to do something useful with that? As usual: it depends.

Why would you use FPGA for generating 4k content