Particularly, where can I browse the source code for any android source application(e.g.the contact application) ? Is the only way to go to download all there is?
Everything is mirrored on omapzoom.org. Some of the code is also mirrored on github.
Contacts is here for example.
Since December 2019, you can use the new official public code search tool for AOSP: cs.android.com. There's also the Android official source browser (based on Gitiles) has a web view of many of the different parts that make up android. Some of the projects (such as Kernel) have been removed and it now only points you to clonable git repositories.
To get all the code locally, you can use the repo helper program, or you can just clone individual repositories.
And others: