You wouldn't, normally. A HMM is used to model sequences of observations, and it would not make sense to use it for image recognition. Unless they are sequential, such as strokes in handwriting.
HMMs are typically used in fields such as speech recognition and part-of-speech tagging. Here you observe a sequence of events that you want to fit to a model in ...
Kalman filter is what you're looking for.
According to Wikipedia:
The Kalman filter may be regarded as analogous to the hidden Markov model, with the key difference that the hidden state variables take values in a continuous space (as opposed to a discrete state space as in the hidden Markov model).
I think a HMM is overkill for this problem. You kind of have 'hidden' states, but they are very limited and dependent on the full sequence of previous states, which you probably want to avoid to make best use of the HMM's features. It also, as you rightly say, leads to a proliferation in states: each dictionary item adds as many states as it has letters to ...
Yes, you can fit any time series (with or without external variables) using HMM, but there are some constraints:
It should follow the Markov property.
There is some variance that other models are not able to capture (in other words, the system is partially observable).
Adding to point 1, for HMM, it should hold true, but the way Baum Welch algorithm ...