Could an FPGA be connected via DMI to a processor, if so how would this be done?
Yes, but it wouldn't be practical IMO. Electrically, DMI (at least of 2.0) was said to be treated electrically as PCIe Gen 2.0 by board designers / hardware engineers. So, any FPGA with transceivers fast enough (Cyclone IV GX or better, Artix-7 w/ 6.25Gbps GTPs or better) could probably chat electrically with the CPU.

That said, the impractical part comes from the fact that the details of the protocol and low-level behavior are likely in red-covered books at Intel that don't leave, unless you're a giant OEM (HP, Dell, etc.) or potentially a BIOS company (Phoenix, AMI).

Remember that the PCH also hosts the BIOS memory (SPI Flash) that the processor executes its first steps from, LPC bus, etc. There's a lot of low-level, trade secret type activity that goes on inside that chip (the PCH) in addition to its publicized functions. You'd need a large, fast FPGA (Virtex/Stratix++) to be able to pull this off, but again, I don't think it is practical.