In answer to your first question: No
Searching for the definition of misplaced modifier on Google leads to the following:
a phrase or clause placed awkwardly in a sentence so that it appears to modify or refer to an unintended word
Similarly, a search on Purdue leads to the following definition:
Misplaced modifiers occur when the subject of the modifier is unclear because the modifier is poorly placed
In this case, the phrase "of Johnsons" clearly modifies the phrase "The family". This leads to the following correct interpretation that the family is composed of Johnsons. So the phrase "of Johnsons" isn't misplaced in the strict definition.
In answer to your second question, the usage is fine.
This is because the sentence is about a family of Johnsons. The word "They" refers to that plurality.
The word "They" actually refers to the fact that a family is composed out of several individuals (hence the plurality). Thanks Kris for pointing that out.