Are there similar equivalences with other HMM algorithms for different questions? A Viterbi-like algorithm for the most likely path, for example, distinct from the series of Kalman-update argmaxes?
so yeah, you can do things like e-m to learn the parameters of the observation and state update models from data (just like fitting an hmm in the fully observed case), or given state and observation models, and past observations you can predict future state given current state. (what it was made for, where state is the state of your rocket given the physics of motion and all the noisy sensors or whatever, which if you think about it is like a viterbi search for the best fitting path through the continuous state space)
The dynamics model in a KF is a linear system with an input and and output (often denoted u and y respectively).
A Hidden Markov Model typically does not have an input. There is an initial state, and then the system transitions "autonomously" (that's a term I'm borrowing from differential equations).
An "Markov model" with input is called a Markov Decision Process (MDP). And if the state of the system is not fully observable, then it's a POMDP.
So Kalman filters are most analogous to "belief updates" in POMDPs. The KF takes into account the known input to the system.