Multiple harmonic star sums involving central binomial coefficients as weights

We consider weighted analogues of multiple harmonic and multiple $t$-harmonic star sums, where the "star" indicates non-strict inequalities between the summation indices. In the special case where all exponents are 2, which is of particular interest also for ordinary multiple ($t$-)harmonic star sums, we obtain evaluations in terms of simple finite sums. The proofs are based on several combinatorial identities which are of interest in their own right.