How can I stop this indexing or whatever this startup process? What is it for? Why is it doing it? Can I defer the loading of such huge process later on when I most need it?
You can actually cancel the build on start. I find this build to be pointless for a variety of reasons:
The way to cancel the build is just simply click on the little red x at the task at the bottom.
When you ready to build then you can just click on run again.
I consider this a bug because it's indeed pointless since it does not make consequent builds any faster nor it is required for the IDE to work unlike what the accepted answer suggests.
However mind that this does not the same as "Indexing" which is Android Studio indeed need. However on a normal startup case you can just cancel the build.
I have an open request for this at Google, we will see if they actually do something with it in the future since I don't see the point of wasting sometimes minutes for a initial build I cannot even use later.