Without photos I can't speak for that one. That one may just be a barnyard mix - but with no photos it's impossible to tell. This one is an EE though. (not a first gen cross if it lays brown eggs)
Yep, no worries.
The bird pictured is an EE. EEs can lay brown eggs.
There is no such thing, EEs are mutts with a chance to lay colored eggs. (meaning they have Ameraucana, Legbar, or Araucana background.)