Let $ A = KQ / I $ to be a quiver algebra with the following two properties:

a) Q is an acyclic quiver

(b) the injective envelope of $ A $ is projective.

Question 1: Is there an algebra $ A $ with properties a) and b) such that $ A $ is not derived equivalent to its opposite algebra $ A ^ {op} $?

(In case you know of simple examples, it would also be interesting to see an algebra $ A $ non derivative equivalent to $ A ^ {op} $ with just the goods a) or b))

Question 2: Does each algebra of Nakayama $ A $ derivative derived from $ A ^ {op} $?

Note that at https://www.sciencedirect.com/science/article/pii/S0021869315000575, at least the singularity categories are equivalent.