What are the limitations, if any, of free EDA tools with respect to high speed design?
There is nothing wrong with a using a tool like kicad to do high speed design. In the end all that matters is that the design and gerbers are drawn correctly to give the correct impedance and lengths. This can be done by hand calculations estimation, PCB calculators, ect. 

What does matter is the time that it will take to get the same result. When a tool offers mostly automatic versions of differential, impedance matching ect, it will save you time. If your time is worth money, then invest in a better tool.

This is also like asking, which is a better tool for making a mechanical piston with a 0.0001" tolerance, a CNC machine and lathe or sandpaper and a file? The same result can be achieved from both, but one will be much easier\costly and one will take much longer\cheaper.

Oh, and don't use allegro, not worth your time. Altium saves time.