During the 1600s, Royal Woods was simply known as Woods.[1] This was where the Loud family's ancestors settled after they were (unknowingly) betrayed by their castle caretaker in Scotland, Aggie, when she banished them from their kingdom.
Sometime in the 400 years between the ancestors' arrival and the present (more specifically during the 19th century), Woods was renamed to Royal Woods, after an oak tree that lived there.
"Driving Miss Hazy" confirms that Royal Woods is in the state of Michigan, and the city is based on Chris Savino's hometown of Royal Oak, Michigan, which is about twenty to thirty minutes away from Detroit, and is considered a suburb of that city.
Royal Oak, Michigan has a very low crime rate, and is considered to be one of the safest cities in America. Since Royal Woods is based on that place, it has the same crime rate. This is mentioned in "Lock 'n' Loud". Most of the crimes that are committed there are minor ones like littering.