expected value of a time series/ stochastic process

I am new to time-series analysis and I found myself getting confused by the most fundamental concepts.

A stochastic process is a collection of random variables $$X_t$$ for $$t=1,\dots,n$$ where each $$X_t$$ is a random variable with its own probability distribution. A stochastic process can generate an infinite collection of realizations, and we call this collection an ensemble. (And an observed time series is 1 realization from that ensemble.) Am I correct in this understanding?

So when we say the expected value of a time series $$E(X_t)=\mu_X(t)$$, do we mean the expected value of a particular $$X_t$$ from the sequence, let's say $$X_3$$, which we can find by calculating the weighted average/integral of the probability distribution of $$X_3$$? And is it correct to assume that each $$X_t$$ in the sequence would have a different expected value since they most likely will not have the same probability distribution?

Similarly, a white noise $$\{w_1,\dots,w_n\}$$ is defined as a collection of random variables with mean 0 and s.d. $$\sigma_w$$. Does this mean each $$w_i\in\{w_1,\dots,w_n\}$$ is a random variable that has a probability distribution with mean 0 and every $$w_i$$ have the same s.d. $$\sigma_w$$?

You're correct in your understanding. A realization is an observation from the infinite ensemble, each $$X_t$$ has its own distribution and at the same time sets of $$X_t$$ have joint distributions. The mean process $$\mu_x(t)$$ describes the mean of a specific $$X_t$$, e.g. $$\mu_x(3)$$ is the mean of $$X_3$$, which can be found by the distribution of $$X_3$$. Each $$X_t$$ may or may not have different means or other statistics based on the stationarity properties of the underlying random process. For example, if the process is mean-stationary, $$\mu_x(t)=\mu$$, i.e. constant, and each $$X_t$$ will have the same mean. But yes, in general, means are different. White noise is an example of a mean and covariance stationary process.