I've never used the STM32F4 series LCD controllers. I don't know exactly what can be done with them. XGA resolution (1024x768) in 24-bit RGB would require a 2.25 MBytes framebuffer, and surely the STM32F4 uCs don't have that kind of embedded RAM to begin with.
Again, look at the PIC32MZ DA - it has 32MBytes onboard specifically for this sort of application.
One option if you still want this kind of architecture with less dependency on supply and more flexibility on specs would be to use an FPGA for the HDMI part, and either write your own controller or use existing IPs.
Yes - Spartan6 will do HDMI natively. You could use the Wiznet W5500 to simplify the networking side of things.
You'd still have to add RAM for a framebuffer, though if it was purely text you may be able to renderfont data on the fly to avoid the need for a full pixel buffer