@ adylse: Job prospects for economic history are what you make them. If you did economics at undergrad you might have a better shot at investment banks. I doubt course choices make much difference (if any) in recruiting.
@ The Mark: Don't know anyone in International Relations.
@ Imbealkariel: You will find a lot of the students have some pretty impressive work experience. My program has people who came from private equity houses, the World Bank, major investment banks, the UN, etc. There are also people such as myself who came directly from undergrad but got some quite decent experience in their field during that time. An undergrad with no experience that is relative to their program will have a lousy personal statement and probably will have a low probability of being accepted.
Interaction between departments? I don't really know, I haven't been looking for it particularly. You're allowed to take electives in any department you want, if that's what you are interested in. Who the professors talk to or collaborate with and the departments they are from is their own business. Polisci and econ collaboration would be like torture for the polisci people because they would have nowhere near the required math background for LSE economics. It is still a mystery to me how political science people "engage" in discussions about economics with only the most superficial of backgrounds in the discipline There seems to be more funding for commonwealth and EU students, yes... feel free to fact-check that though.
Academic rigor... well that depends on your department, what you're good at, and how strong and well-suited your background is for your LSE program. There is a wide variety of standards for undergraduate education around the globe, and within America as well. Among my peers here, I've met some impressive students from Hamilton College, Brown, Harvard and UC Berkeley, who all seem to do very well with the material. I've also met students from other schools such as University of Pennsylvania leave me wondering what the heck they are teaching (and who they are admitting) at supposedly top US universities. I don't think any program at LSE is easy, because even when the subject matter is relatively more straight-forward, the expectations of the work you produce are always very high at the postgraduate level. IR and development studies are areas in which I have very little interest so really can't give you much proper insight there.