I like your question but don't have specific answers to 2 and 3? I imagine that software packages like
SAS (broadly speaking of SAS products and not just SAS/STAT) may have tools that facilitate simulation but I can't say for certain. I don't think this sort of thing fits as a branch of mathematics or statistics.
Now question 1 is what I would like to focus on. Simulation can help in learning statistics at all levels and can aid in statistical research in general. Indeed, there are journals focused on simulation and computation. Even the FDA is recognizing the importance of simulation in designing clinical trials and helping predict outcomes.
In the 1960s Julian Simon taught introductory statistics using simulation as a motivator. Although controversial he later claimed that he was doing resampling (permutation and bootstrap) prior to Efron. He published a book using these ideas in 1969. It certainly lacked the theory and was only a teaching aid and not a new approach to statistical estimation. He did not develop any of the mathematical properties that came with and after Efron.
I think for introductory statistics it is useful to do a simulation to demonstrate sampling distributions, and show how the central limit theorem comes about and physical simulation through the quincunx demonstrates the DeMoivre - Laplace version of the central limit theorem.
Sometimes it enhances intuition. I think that the Monty Hall problem is puzzling and seemingly paradoxical even to mathematicians like Paul Erdos. But simulating the game is often very convincing. There are many problems in probability theory that are counterintuitive and simulation can, I think help.
In 1978 when I was working on my PhD in extreme value theory I had an intuitive idea for a limit theorem that I was trying to prove. I struggled with mathematics. Then I decided to simulate the stochastic process and the simulation "confirmed" my result. This gave me the confidence to push on a prove it.
So even at the graduate level and beyond simulation can be useful in two ways.
To help develop intuition as you suggested in question 1 but also
To confirm intuition as I did in my thesis