Compilation is a process to optimize file access on php classes in magento.
In a usual environment you should not activate it, as the php solutions like opcache solve this problem perfectly with less problems and side effect.
caches are what the name says, parts which get saved (mostly for a limited time) for the next run, because its unlikely they will return a different result.
index management is something similar to compilation, but not for php classes, but for database data.
While magento is possible to work without the flat tables, there is no alternate way for the other indexes, so they should be up to date, or you will have old data.
Here it is says, they have 2 modes, update on save, and manual update.
For update on save you should have no problems as long the modules are all cleanly build.