How to combine a perfect signal with a limited dynamic range with a poor one with high dynamic range?
In response to your question (1):

Thinking along the lines of your initial idea, there is a way to avoid regression and do the following approximate procedure instead: 

Without loss of generality, assume that $\epsilon$ has zero mean (if not, then just absorb it into the constant $\beta_0$). Also assume it is i.i.d and has finite variance. Suppose you obtain two sets of data for $(f,g)$ pairs when $h=1$. I will denote these as $\{(f_{1,i}, g_{1,i})\}_{i=1}^N$ and $\{(f_{2,i}, g_{2,i})\}_{i=1}^N$.

Let $\bar{f_1} = \frac{1}{N} \sum_{i=1}^N f_{1,i}$ and similarly define $\bar{g}_1$, $\bar{f}_2$ and $\bar{g}_2$. If $N$ is sufficiently large, we can now claim that, at least approximately, the noise $\epsilon$ can be averaged out and we can write:
$$
\bar{g}_1 \approx \beta_0 + \beta_1 \bar{f_1}
$$
and
$$
\bar{g}_2 \approx \beta_0 + \beta_1 \bar{f_2}.
$$
Now we only need to solve two equations in two unknowns to get the scale parameter $\beta_1$ and offset $\beta_0$, provided $|\bar{f}_1 - \bar{f}_2|&gt;0$.

Of course, whether this will work in your application will depend on whether the variance of $\epsilon$ is small enough (remember: averaging will reduce the variance by a factor of $1/N$), and whether the $f$'s are really noise free when $h=1$.

In response to question (3):

The same procedure should work in case $f$ is a quantized version of the true velocity. The argument here is that quantization introduces a small amount of noise that is approximately uniformly distributed with zero mean. It gets averaged out for large $N$, or so I hope.