I know that Markov chains that are irreducible and aperiodic are guaranteed to converge and have an invariant distribution, but can a non-irreducible one do too? If so, what would be an example?
Also, is there a different between an invariant/stationary distribution and a convergence for MC?
Concerning your first question:
Just take an irreducible aperiodic chain and add a transient state from which you can only "flow" into the states of the irreducible chain (which then form a recurrent, aperiodic communication class). The so obtained chain is reducible.
Concerning your second question:
Yes, there is a difference. There are Markov chains with several stationary distributions where the initial distribution decides where the chain converges to. And there are Markov chains with only one stationary distribution but where convergence depends heavily on the initial distribution (for the last satement see for example here p. 31).