I am trying to learn something about Bayesian statistics, however, I am struggling already with the simplest equations and, moreover, with the very basic questions: What are we given? What is our goal?

As far as I understood the most basic setup we are given a probability space $(\Omega, \mathcal{A}, P)$, a space with $\sigma$-algebra $(V, \mathcal{B})$ and a random variable $X : \Omega \to V$ which has a density parametrized by $\lambda$, $f(x;\lambda)$ and we also assume that $\lambda$ itself is drawn from a random variable $\Lambda : \Omega \to \mathbb{R}$. Moreover we are given an observation $x$ and try to figure out something about $\lambda$ from this observation.

Then people usually write down something like $$p(\lambda|x) = \frac{p(x|\lambda) p(\lambda)}{p(x)}$$

The expressions above are then evaluated to

$$p(x|\lambda) = f(x;\lambda)$$ $$p(\lambda) = g(\lambda)$$

Question 1: What is the meaning of '$p$'? I can hardly believe that these are probabilities, because although one could give $P(X=x|\Lambda=\lambda)$ a useful meaning [as the factorization of the conditional expected function $E(1_{X = x} | \sigma(\Lambda))$], the symbol $p(\lambda|x) = P(\Lambda=\lambda|X=x)$ would somehow still be zero all the time (as $\Lambda$ is continuously valued). So I assume that these are supposed to be densities.

Question 2: What is $X$? Isnt it merely true that we have a function $\tilde{X} : \Omega \times \mathbb{R} \to \mathbb{R}$ [not even jointly measurable] with the property that for every $\lambda \in \mathbb{R}$ fixed, $X_\lambda := \tilde{X}(\cdot, \lambda)$ is a random variable with density $f(\cdot; \lambda)$? Or are we considering $X(\omega) = \tilde{X}(\omega, \Lambda(\omega))$? If so, why is this measurable?

Question 3: If $p(\cdot)$ is supposed to be a density, what random variables do we put inside $p(x|\lambda)$? I assume that its supposed to be the conditional density of $X_\lambda$ and $\Lambda$. But here is the question: can one define conditional density of two random variables without the two random variables being jointly distributed? I know that one can put $f_Y(y|X=x) = \frac{f_{(X, Y)}(x,y)}{f_X(x)}$ but are the two random variables $X_\lambda$ and $\Lambda$ jointly distributed? This seems like an unnatural assumption...

Question 4: I guess that even if one can define conditional densities of random variables even if they are not jointly distributed then $p(x|\lambda)$ is some kind of abstract definition, so why is that the simple term $f(x;\lambda)$?

I am sorry if these are very basic questions but all the resources I was able to find skip over all these details, so pointers are very welcome :-)