Well, LO feedthrough is just however much RBW you have. You necessarily* get a spike at 0Hz offset.
*Actually, I suppose one could make a spec with a very carefully balanced mixer, that rejects the LO signal, and perhaps also which detects signals orthogonal to it. Then 0Hz would be very attenuated or entirely invisible, and signals down to 0Hz (within uncertainty of the RBW) would be resolvable.
As for standards, I believe some CISPR and MIL standards start at 9kHz, or something similar to it, thus demanding instruments start there. (Most domestic standards, i.e. FCC Part 15 and CISPR 22, cut off at 150kHz.)
I've seen at least one R&S analyzer that claimed something like 25Hz to 10s of GHz. It was safely mounted in a rack, and I'm pretty sure that, if I had merely touched it, that fingerprint alone would've depreciated it by at least my wages that day. (Didn't see it in operation though.)
I believe, some of the more stringent MIL standards (and probably going into SIGINT territory as well) place limits on conducted and radiated emissions over an extremely broad range, i.e., anything from power supply ripple in the 20Hz range, up to radiated noise in the GHz. The LF limits might be quite modest for most equipment (easily met without special precautions), but I don't really know how tight the limits go (especially for security roles).
Tim