Are there any effects from breathing the fumes for too long from a decomposing human body; can you catch anything?
Possibly. While the "fumes" of decomposition, which are really gasses created by a variety of bacteria, are often unpleasant, these gases are probably not harmful in and of themselves. However if the corpse was harboring one of a number of spore-transmitted diseases, it is possible to become ill from one of those. There are a couple of recorded cases of tuberculosis being transmitted from a corpse to those handling it, including one where it was found that the act of pumping embalming fluid into the body displaced TB spore-laden air from the corpse's respiratory system, creating a real danger to the mortuary workers.
Aside from tuberculosis, anthrax is another disease that can form highly resistive spores that could be transferred from a corpse to workers nearby. It it believed that anthrax spores can even survive and be lofted into the air from open funereal pyres. Hepatits is also suspected to remain transmissible for some time after death.