Well, you definitely could feel sick from the foul smells coming from a dead animal. You could even physically throw up. If you mean from airborne bacteria or viruses from smelling a dead animal, you likely would not ingest enough to have an effect.
A opposum will play dead, but I have never known them to smell rotten. In fact, playing dead doesn't always work out to their advantage. I have many possums in my are, and everytime my dogs encounter them, they try playing dead, but my dogs aren't fooled. Poor possums.
because all the smells are on the dog dumbo Some dogs will roll in things when they are outside such as cow manure, dead animals etc. They think it smells great. Of course you probably don't agree when the dog comes in the house!
There could be a dead person or dead animal in the house, or spoiling meat as from a refrigerator or freezer. Vultures are attracted by the smell of carrion and many species have an exceptional sense of smell.
The planet Mercury smells like refried beans. Venus smells like gasoline. Earth smells like unwashed socks. Mars smells like Martians. Jupiter smells like muddy dogs. Saturn smells like burnt paper. Uranus smells like dead trees. Neptune smells like a turtle. Pluto smells like squab.