In the event of a disaster at any nuclear power plant other than Chernobyl, would we be dealing with the emission of radioactive iodine isotopes? Would any nuclear catastrophe emit radioactive iodine isotopes?
Do some power plants have different cores that would emit other radioactive isotopes but not iodine isotopes?
 A: Yes. Iodine is a common product of fission reactions, whether the fuel is uranium, plutonium, or thorium.
In nuclear fission, each fuel nucleus splits into two smaller nuclei (or sometimes more than two), and releases one or more neutrons. The product nuclei usually have an excess of neutrons, which makes them radioactive, and they decay by beta emission, often accompanied by gamma emission.
Here's a plot from Wikipedia of products for the common fission fuels. The horizontal axis is atomic mass. As you can see, a wide range of products are possible.


Fission product yields by mass for thermal neutron fission of U-235 and Pu-239 (the two typical of current nuclear power reactors) and U-233 (used in the thorium cycle)

Please see Nuclear fission product for further information.
All isotopes of an element have virtually the same chemical properties as each other. The only difference is the reaction rate, which is a function of the isotope mass: heavier isotopes react slower. The difference in reaction rate is important for light elements, but it's negligible for heavier elements.
Iodine is an important trace element. The body stores iodine in the thyroid,  but the thyroid has a limited capacity, and excess iodine is quickly eliminated from the body. Having a small excess of iodine isn't a health hazard. If someone with an iodine deficiency is exposed to radioactive iodine, it will be quickly concentrated in the small thyroid body. However, if the thyroid is already saturated with iodine, virtually none of the radioactive iodine will be stored. So giving iodine supplements is an excellent preventative measure against radioactive iodine.
The two most important radioisotopes of iodine in radioactive fallout from reactor accidents and weapons are iodine-129, which has a half-life 15.7 million years, and iodine-131, with a half-life of eight days. Generally, with all radioisotopes, short half life isotopes have the most energetic radiation, and their daughter isotopes are also likely to be radioactive.

Another important element in fission reactor accidents (and bomb fallout) is strontium, especially strontium-90, which has a half-life of 28.8 years. Strontium is chemically similar to calcium, and it affects biological systems that normally use calcium. So strontium gets concentrated into bones and bone marrow, and teeth, especially in children. It also finds its way into milk. As far as I know, shielding against strontium via a high calcium diet is useful, but probably not as effective as the process mentioned above for iodine.
From
https://remm.hhs.gov/calcium.htm

Calcium is an alkaline earth metal, as are strontium, barium, and radium.
A mass effect from calcium can interfere with absorption of the other alkaline earth metals and compete for bone deposition. In the event of internal contamination with radioactive strontium (Sr-90) or radium (Ra-226), generous doses of calcium preparations may be beneficial.

From PubMed, Prevention of accumulation of strontium-90 in the bones by fortifying the food with fish products, V A Knizhnikov et al. Vopr Pitan. (1991)

The replacement of 1/5 part of the usual ration received by non-inbred white rats for fish mass containing up to 4 g/kg Ca resulted in a sharp (by 70%) reduction of Sr-90 accumulation after its oral intake.


Another dangerous radioisotope that's mentioned in connection to reactor accidents is caesium. Caesium has 40 known isotopes, and only caesium-133 is stable. The most important one in fallout is caesium-137, with a half-life of 30 years. Caesium is in the same chemical family as sodium and potassium (the alkali metals), which are important in many biological processes, especially in the nervous system. The ions of these elements are highly water-soluble, and have a fairly short biological half-life: around 11 days for sodium, 30 days for potassium, and 70 days for caesium (closer to 50 days in children). As Wikipedia mentions, caesium has a tendency to accumulate in the pancreas.

Bandazhevsky found a concentration of ${}^{137}\rm{Cs}$ 40-45 times higher than in their liver, thus demonstrating that pancreatic tissue is a strong accumulator and secretor in the intestine of radioactive cesium

However, the worst accident involving caesium-137 was not caused by a reactor or a bomb, it was caused by the ignorant handling of a stolen Cs-137 radiotherapy source. Please see Goiânia accident for details.
