At least when the group is simple (or semisimple), irreducible representations are classified precisely by their highest weights and classical formulas for dimensions and tensor product multiplicities apply. (The group can't be a torus, so in the remaining reductive cases one has to factor in a power of det or other such character to get all irreducible representations.)
Concerning property 1, you have to have a group whose 3-dimensional irreducible representations all fail to be self-dual (ruling out for instance $\mathrm{SU}(2)$): the multiplicitiy of the trivial module can be computed by the dimension of the Hom space from it to the tensor product in question, which is easy to analyze.
It helps to give one or more starting examples where both properties hold. One of these would be $\mathrm{SU}(3)$, which has just a single irreducible 3-dimensional representation (the natural one) and its non-isomorphic dual. (Allowing a reductive group with a nontrivial torus as center will complicate things a bit.) Beyond rank 2 the 3-dimensional irreducibles seem to be very sparse, so I don't immediately see any further examples among the simple or semisimple groups. Weyl's dimension formula is a natural starting point here.
P.S. I'm not sure about the exact setting of the question (or its motivation). For instance I'm taking for granted that irreducible representations are over $\mathbb{C}$.