Is it possible for the previous version android module(or any other) to work with new unity version?

Or is it necessary to download them again every time?

 

Yes, it's necessary to download them again every time. That said don't feel pressured to move to new releases. You can stay on an older release for as long as you want (there are only a handful of situations that would force you to upgrade like support for a new platform or something similar but they're rare occurrences).