{Lavinia of the Tenth} is being cast, opponent casts {Essence Backlash}, does prot. red protect against it?
Protection only functions on the battlefield. Red counterspells can counter her perfectly fine.
Fair enough.
But would a creature with "Protection from red" be protected from for instance a creature that is {R}{U} ?
(Basically, is that dual-colored creature considered to be both colors individually ?)
Yes. A {u} {r} creature is both blue and red, and so for example couldn't block a {dryad militant} with {brave the elements} for red OR blue.
When any spell or ability checks for a condition, it only cares about that condition and will ignore any others. In this particular case, it only cares if the object is red the fact that it is also blue doesn't matter to the creature with protection.