I have a question about the "hand-wavy" definition of these terms, or just whatever helps to form my intuition.
When I encounter terms like $\int_0^t f(s) dW_s$ for $f(s)$ deterministic function, I think I know what it means. It is well-defined for $f$ a constant function (in which case the result is a Normal distribution with mean 0 and a certain variance). Then $f$ can be approximated by dividing the interval $[0,t]$ into $n$ rectangles, and that is well defined too. Then the resulting integral is the limit of that approximation as $n \to \infty$.
When I encounter terms like $\int_0^t W^3_s dW_s$ then we can evaluate it using Ito's Lemma, in which case the answer may still contain more integrals in the form of $\int_0^t f(s) dW_s$, but either way they're well defined, and the final result is still a distribution.
But when I encounter terms like $\int_0^t W_s ds$ or $\int_0^t e^{W_s} ds$, what does it mean? I've seen in various stack exchange questions that it's a distribution and you can compute expected value and variance, but is it a Normal distribution? Moreover, what does the integral mean?
Is this interpretation correct: that you (in principle) take $n$ realized path of the Wiener process, and for each path, evaluate $\int e^{f(s)} ds$ where $f(s)$ is a particular path. Then you take the average of those $n$ paths, and as $n \to \infty$ your average should converge to the $\int_0^t e^{W_s} ds$. Is this right? First of all, it's hard to see that it converges at all. But second, then the answer I get is a number, and that's the mean of that number. I still don't have the distribution.
Thank you