Dims aren’t unique to timeseries, they’re a way to organize you data and make your code more readable. See here for a full explanation.
init_dist gives the initial conditions for the recursive relationship. For example, in the stochastic volatility model, the variance of the time series follows a Gaussian random walk, so we write the model as:
\begin{align} \sigma_{t+1} &= \sigma_t + \eta_t, \quad \eta_t \sim N(0, \sigma_{\eta}) \\ r_t &\sim T(0, \sigma_t, \nu)\end{align}
This model isn’t closed because you need to tell me what \sigma_0 is. In our case it’s a parameter to estimate, so we need to specify \sigma_0 \sim f(\cdot) where f(\cdot) is the init_dist.