You can actually see this in real-time if you go into the developer mode of your browser. The script is running in real-time and the animation is still running in the browser window. So you see the outcome of the bet in the script that are running, before the output is send to the browser.
There is not a long delay between the script running and the animation being done in the GUI interface, but you can see the result, before it is displayed in the browser. (Just nice to see the clock work of the software being run in the background for the people who are interested in that)
That's correct, there are some ways to check the results faster, but why would you do that? I feel like you could also refresh the whole page right after betting and your balance would update faster.
I never bother to do it, I'm not that impatient.
Animation isn't annoying in these games and doesn't take forever. It's actually there for your pleasure, so that you can see the balls landing in certain spots, even though it's calculated instantly. I also don't try to skip slot animations because they have a certain rhythm.