The task is to filter x(t) when given y(t), where y(t) = x(t) + n(t).
Great but first we need to build an appropriate filter. At this point:
No. The task is to filter $y[n]$ to achieve $x[n]$. Your filter acts on the available signal $y[n]$ to get a best estimate of $x[n]$ from a noisy observation of it. BTW. discrete-time uses $y[n]$ or at least $y(n)$ notation, and not $y(t)$. In DSP, $t$ is reserved for continuous-time.
1) Are we assuming that we have access to some training samples of
x(t) and/or n(t) in addition to y(t)?
trainig samples is a term related with adaptivity, pattern recognition, neural networks etc. In the classical Wiener context, you do not need training samples but you need data statistics. See answer 2 for how to obtain the statistics.
2) Or are we just assuming that we have access to statistical
properties(auto correlation, cross correlation) of x(t) and/or n(t) in
addition to y(t)? If so since x(t),n(t) are not explicitly available,
are these statistics computed from other signals available to us with
similar statistical properties?
Yes. In the Wiener filter context, you either assume (for theoretical development) or estimate (for practical applcations) the correlations (or Power Spectral Densities) between all those signals. In practice you estimate them from avaiable data by several means, depending on the application. You need data for this; and that's probably what you call training samples.
3) Or is it a combination of both, such as having access to samples of
x(t), but having only access to second order statistics of n(t) (or
You do not have access to relevant noise-free $x[n]$ by any means; otherwise why filter $y[n]$ to get $x[n]$? But, in some cases, one can have noise free irrelevant blocks of $x[n]$ samples, in addition to the relevant noisy blocks. So you can use those noise free blocks to estimate certain parameters of the process $x[n]$, so as to filter out the noisy blocks.
4) Is the noise n(t) always assumed to be uncorrelated to x(t) in
Wiener filtering, or are the textbooks going for the simplified
version? What is the default scenario?
Yes. The noise $n(t)$ (or better $v[n]$ in discrete-time context) is almost always modeled as additive, white, zero-mean Gaussian noise uncorrelated with $x[n]$. This simplifies all the mathematical development and in typical cases is a good model. But you can relax some of those assumption, if that would yield a better (but more complex to derive) filter...
5) Why are all the examples coming up with all-zero filters but not
using pole-zero filters? Is it just a matter of simplified textbook
treatment or is the problem definition not amenable to using transfer
function modelling approach? Yes the first approach is computationally
easier, but these days we have significant computing power and I don't
think it is a good justification anymore, if that's the only reason.
IIR filters are good for their computational efficiency (and in Wiener context, they also yield the minimum MSE compared to FIR) but they may suffer from unstability and causality problems, and their design and analysis is more complex in adaptive filtering context. For this reasons, it's simpler and safer to use FIR based filters for adaptive Wiener applications.
To learn more about the Wiener filters and adaptive Wiener filtering: