Silicon controlled rectifier P-type gate connection and layer doping
You do have to be careful, there are about 13 different devices that are classified as being SCR's.

1) You can have NPNP or PNPN structures for thyristors, so if you build the complementary structure your gate is also complementary. But for a given structure , no you can't change that as the injection of current has to occur within a certain region.

2) Well only one of the "inner" layers would be considered to be lightly doped going from cathode to anode in one structure we would have cathode N+ @ 1e20, gate P @ 1e18, n- @ 1e14 and anode P @ 1e19.  Of equal importance are spacings in the structure.

In reverse bias mode most of the voltage is supported by the thick lightly doped n- region until punch through or avalanche causes conduction.