1. So, I actually looked this up. The energy required for induced radioactivity depends on the element being bombarded, but for the production of most dangerous isotopes, you're looking at something in the 10 MeV range. Classically speaking, these would be considered gamma rays, and quite powerful ones too.
However, this is complicated by the fact that terminology has shifted. In medicine and some other fields, the split between X-rays and gamma rays is one of source, not power. Some types of specialized radiotherapy equipment can generate "X-rays" of up to 25 MeV - absolutely enough to cause induced radioactivity, which is why they're so specialized. (It should be noted that other fields, chiefly astrophysics, continue to divide X-rays from gamma rays based on their frequency and power.)
Interestingly, rerevisionist is wrong about neutron augmentation (I'm assuming he means neutron capture) being the mechanism - what happens in gamma-ray-induced radioactivity is that the gamma ray is strong enough to actually "knock" a particle (usually a neutron, sometimes a proton) out of the nucleus, making it unstable. (It's also possible for gamma rays to straight-up induce fission, but only in nuclei that are already unstable.)
So basically: yes, it is possible for electromagnetic radiation to induce radioactivity, and it can happen to elements not normally considered radioactive. However, it requires very high-energy particles, which most non-specialists would know as gamma rays, not X-rays. (So the X-ray machine in your doctor's office, for instance, isn't going to be creating any weapons-grade uranium any time soon. Thankfully.) I also give no estimate as to how much radioactive material you could generate using this process; certainly its medical and scientific applications are aimed at small volumes of difficult-to-obtain products.
2. The salient question is of course "how". Firebombing can level a city, to be sure, but is a slow process. Conventional explosives mimicking the effects of nuclear detonations are possible, but limited in scope. The largest known to have occurred - the US "Minor Scale" test - simulated a 8 kT (half the Hiroshima bomb) detonation. No one seems to have recorded how big it was (although the bomb, if you can call it that, weighed 4744 (short) tons), but another conventional mimic test, Operation Sailor Hat, simulated a 1 kT detonation with a solid half-sphere bomb 17 feet high and 34 feet across! Anyone proposing that Hiroshima was conventionally mimicked is invited to give their own explanation for how the US could move a 20,000-cubic-foot explosive, weighing between 8 and 10 tons, into Hiroshima without anyone noticing.
3. The neutron bomb is technically a thermonuclear fusion device. A standard fusion warhead ("H-bomb", although you can use helium too) is a three-stage device. The first (fission) stage produces the heat and pressure needed for the second (fusion) stage, which does indeed produce a flood of neutrons among it byproducts. These neutrons are then used to induce fission in the bomb casing (which is mostly depleted uranium), adding more energy to the detonation and thus enhancing yield. In a neutron bomb, this element is left out (there are some other changes too, but this is the major one), so that these neutrons pass into the environment.
Obviously the final stage of a standard fusion warhead, which creates tons of uranium fission byproducts and then flings them into the air as hard as it can, means that the fusion bomb is not noticeably "cleaner" than a fission one. (It might be cleaner than a fission device of the same yield, but since a fission device of the same yield would be the size of a house, this isn't of much help.) It's also worth pointing out that neutron bombs aren't exactly great for the environment; neutron activation of steel and other components in nuclear plants is by some measures more of a contamination problem than the fuel itself.
The neutron bomb also doesn't, contrary to popular belief, "kill people and leave buildings standing". It kills people and buildings equally. (It still has a shockwave, thermal pulse, and fireball that are all pretty hefty by the standards of conventionally munitions.) It's actually much more efficient at destroying tanks, though. Remember how a fusion bomb uses a final stage of depleted (or near-depleted) uranium that's touched off by the neutrons? Depleted uranium is extensively used in tank armor (for its unparalleled density and, ironically, gamma ray shielding), which has predictable results when it's flooded with neutrons. So the neutron bomb is most effective at killing the very units that are best protected from "normal" nukes... which means they can be a lot smaller, and therefore less likely to clip civilian targets by accident.
One final point of interest about neutron bombs, which I'm only including because I just found out about this and it fascinates me. If the neutron pulse hits another nuclear weapon, it tends to create partial, premature fission. If a hydrogen bomb's fissile elements are partially fissioned, they can't go off the way they're meant to later, which means they'll almost certainly fail to cause fusion, robbing the bomb of the vast majority of its yield. (To say nothing of what this is doing to the bomb itself, sending it off-course, screwing with the detonators, and whatnot.) Thus one primary role for neutron warheads... is in anti-ballistic-missile missiles.
4. The only wars any of those powers lost were when their various puppet states failed to govern effectively; that's not something you can solve with an atomic bomb. (If you bomb all the people who don't like your puppet state... there won't be much of a state left to puppet. Defeats the whole purpose.) The exception is Israel, which won all of its conflicts actually taking place in Israel; the only land they lost was a chunk of Egypt that they couldn't really keep in the first place and certainly wasn't worth escalating the conflict over.
@Dr. Razark: Presumably they would say those cities were firebombed (a large number of Japanese cities were firebombed during WW2, some very thoroughly; the repeated bombings of Tokyo actually caused more deaths in total than the destruction of Hiroshima) - most of the immediate deaths from an atomic bomb are from the thermal pulse and resulting fires, which if you're conspiratorially minded could easily become "they dropped lots of firebombs and just said it was an atom bomb". The problem is, as I mentioned above, time: Tokyo was reduced over a matter of months, so while it might be just about plausible now to believe that Hiroshima and Nagasaki were firebombed, at the time it would've been very obvious what the difference was.