Can I have a Markov chain where one state doesn't transition to any other state? (but some other states transition into it)
People often consider square matrices with non-negative entries and row sums $\leq 1$ in the context of Markov chains. They are called sub-stochastic. The usual convention is the missing mass $1- \sum [$entries in row $i]$ corresponds to the probability that the Markov chain is "killed" and sent to an imaginary absorbing "cemetery" state, when it is state $i$.
Sub-stochastic matrices often crop up in Markov chain calculations e.g. when we consider only states in a communicating class, or just transitive states etc.
Row sum of transition probability matrix need to be 1 because the states we define should be exhaustive.
Example: There should not be a customer who are not in any of the states defined for the markov model.