Yes, I think too that this notation is a bit unfortunate (exactly as for your question on $\partial/\partial_{t_{\text{coll}}}$).

First, the $t_1$, $t_2$, ... notation used in the article is non-sense. Because you only have one time $t$ and you do not create additional "fake time" as in typical [Multiple-scale pertubation][1] theory. Usually, what is written as:

$$\partial_t g = \partial_{t_0}g + \epsilon \partial_{t_1}g + ...$$

in the article you linked, is written as:

$$ \partial_t g = \partial_t^{(0)} g + \partial_t^{(1)}g + \dots $$

See for example Contemporary Kinetic Theory of Matter (by Kikpatrick) or The Mathematical Theory of Non-uniform Gases (by Chapman itself, available on the Internet Archive). This notation makes clear the fact that the "derivative symbols are not really derivative (although they act a bit like it). Our master Chapman tells it himself (p. 113 or 137 of the PDF with $\partial_r\equiv \partial_t^{(r)})$:
[![enter image description here][2]][2]


I would say it's better to write the derivative this way:

$$\partial_t g = (\partial_t g)_0 + \epsilon 
^1(\partial_{t} g)_1 + \dots,$$

where we assumed that the **result** of a time derivative can just be expanded into a series in power of $\epsilon$. Each terms in $(\dots)_i$ is of order $\epsilon^0$. But they will contribute to a different physical process. The zeroth order term will give you the Euler equation (Eq. 15 in your pd); hence, they correspond to phenomena happening on a time scale similar to the inverse wavelength since these are described by waves or advection ($w=ck$). The first-order term will give you the dissipative contributions (viscosity, heat diffusion, ...) to Navier-Stokes, which happen on a diffusive timescale ($w=Dk^2$). 

This somehow justifies the idea that the author had of writing these things in term of derivative with different "timescales",  as in regular multiscale perturbation (but again, that's just completely non-sense mathematically, and confuses more than it helps). Indeed, $\epsilon$ keeps track of the gradients. The zeroth order correspond to term proportional to single gradient and the first order to term proportional to double gradient (Lalacian) in the transport equations. Hence $\partial_{t_0}$ corresponds to the change in time of some variable due to the single gradients terms, $\partial_{t_1}$ corresponds to the change in time due to double gradients.. and so on

Concerning your last question $f^{(0)}$ is the local equilibrium approximation of the probability distribution (which creates the Euler equation and the single gradient expansion) while $f^{(1)}$ contains terms proportional to the gradient of the macroscopic fields (which create the additional terms necessary to obtain the full Navier-Stokes equation)

  [1]: https://en.wikipedia.org/wiki/Multiple-scale_analysis
  [2]: https://i.sstatic.net/oJpgqYHA.png