Functions with subscripts? In the equation:
$f_\theta(x)=\theta_1x$
Is there a reason that $\theta$ might be a subscript of $f$ and not either a second parameter or left out of the left side of the equation altogether? Does it differ from the following?
$f(x,\theta)=\theta_1x$
(I've been following the Machine Learning class and the instructor uses this notation that I've not seen before)

As you note, this is mostly notational choice.  I might call the $\theta$ a parameter, rather than an independent variable.  That is to say, you are meant to think of $\theta$ as being fixed, and $x$ as varying.  
As an example (though I am not sure of the context you saw this notation), maybe you are interested in describing the collection of functions $$f(x) = x^2+c$$, where $c$ is a real number.  I might call this function $f_c(x)$, so that I can later say that for $c\leq 0$, the function $f_c$ has two real roots, while for $c>0$ the roots are uniformly convex.  I think these statements would be much more opaque if I made them about the function $f(x,c)$.
