Here is a government fact sheet about septic tanks.
http://water.epa.gov/aboutow/owm/upload/2009_06_22_septics_septic_systems_factsheet.pdf
Yes, they are very common in the US. Keep in mind how large the US is and how very diverse our topography is. In many places, the state or local municipality has not engaged with a water/sewer company to run water lines and waste-water lines through the area connecting homes. There are many homes here that are in very rural locations, long distances from waste water treatment plants.
Some of these homes have both well water for their potable water and septic for their waste. Some have municiple water sources but still use septic for waste. In some locations, residents are given the choice to "tie in" to the municiple system (usually at a steep price) or use a septic system when the lines are installed in an area. Some local governments require people to tie in. (Just for understanding septics - there are several scenarios).
There are many different grades and designs of systems but they all work out in the end the same way. The type of system used is dictated by when it was installed, where it is being installed, the ground quality (sandy, clay, rocky, etc), how close it is to protected waterways (creeks and culverts that drain to streams and rivers, etc). The size is usually dictated by whether it will be for public or private use. For a residential home, it is usually based on the number of bedrooms in the home. Most localities count two people for each bedroom to calculate what the use of the system will be. In this instance, if the trailer and the house shared a system, it could be "grandfathered" in. I don't know which came first - the house or the trailer - but in many rural areas, people buy land, place a trailer on it and live in the trailer while they build a house or develop the land, then rent the trailer out.
Yes, in a county such as Pulaski, it would be reasonable for LE to know/assume whether there was a septic system there (especially given that the whole county is rural and neighbors have said everyone there is on septic). That doesn't mean they would know exactly where it is without going to the county records. The location (usually with a little hand drawn map of the property) and permits for the septic system should be on file with the county.
As for solid objects in the tank...the clogs that would back a system up in to a house usually happen in the line. The tank was quite large (in contrast to the size of a child being in it) and if it was not full, it wouldn't back the system up unless the child's body was blocking a pipe that flows in. It might have taken a few more days to back up, so to speak.