The ontology of pure states is tricky, but if you believe in pure states then mixed states are fairly straightforward, I think.
Teleportation of a third qubit seems unnecessary in this thought experiment. The same issue arises if Alice and Bob hold halves of a Bell pair and Alice sends an email to Bob stating the outcome of a measurement she performed on her half. The rest of this answer is about that simpler experiment.
The no-communication theorem means that if Bob is sure that he'll never have access to Alice's qubit again, then he can assume without loss of predictive power that she has already measured and discarded it, and the wavefunction has collapsed to some pure $|\phi\rangle$, but he doesn't know which.
His knowledge can then be represented by a classical Bayesian probability distribution over possible values of $\phi$. This needn't be a uniform distribution—maybe he knows that Alice prefers to measure in the Hadamard basis—but at the very least, unless he thinks that Alice has the power of postselection or might have rigged the qubit creation process, he should believe that $|0\rangle$ and $|1\rangle$ are equally likely, $|-\rangle$ and $|+\rangle$ are equally likely, etc. In that case Bob's beliefs about likely measurement axes have no effect on his prediction of the outcome of any experiment on his qubit. You can conclude this from the no-communication theorem again or by directly calculating the outcomes of measurements on all axes.
A mixed state is just a compact way of representing this "classical knowledge about a pure state modulo distinguishability via experiments on the state". It's updated in light of new information in the same way as an ordinary probability distribution because it's just a representation of an aspect of that distribution.
A mixed state can't replace the full probability distribution for all purposes, only for the purpose of answering questions about experiments on the state. For example, if someone offers to bet Bob that Alice will measure in the Hadamard basis, whether he should take that bet depends on his beliefs about Alice's basis preferences, and the density matrix doesn't contain that information. If Alice sends an email saying she measured her qubit and the result was $1$, but she doesn't say what gates she applied to it first, then Bob's beliefs about the results of experiments on his qubit after getting the email might be represented by a state like $0.9|+\rangle\langle+|\,+\,0.1|-\rangle\langle-|$. This can't be calculated from the mixed state derived from his beliefs prior to getting the email; you have to update his full set of beliefs and calculate a new mixed state from that.