You have managed to pack in a few question in one post, let me try to summarize in simple terms how Realm works so you may decide how you can use it to accomplish your requirement.
Let's say you have two users named Frank and Peter and you have a model class called AccountBalance.java. Peter have a balance property with a value of $5 in his AccountBalance object and Frank has a balance of $10.
Scenario 1 - Stand Alone Realm Database
Realm Database can and do work as a standalone database, each database is private to the app that created it. Peter and Frank will never see each other's account balance. If either of them has two Android devices, the values of the account balance will be different on each device because the databases are different. In this capacity, Realm is a better replacement for SQLite.
Scenario 2 - Synched Realm Database
In this capacity, your local Realm Database is backed by a cloud-based server running Realm Object Server. You setup, install and maintain this server on your own. With this, when Peter performs an operation in your app that changes his account balance from $5 to $3 then that change is first saved locally to the Realm Database on Peters device and also copied over (synched) to the cloud server. If Peter launches your app on another device and logs in with the same account that he used on the first device, then his data will be downloaded (synched) from the server to the new device. At no time will Frank see Peter's account balance and vice versa, unless you implement such feature in your app.
Now if you want to write server-side code that can perform an action after a change happens, then you may want to upgrade to Realm Professional version. There is cost associated with this version and I do not think that you need to consider this option at this time since it appears you need more clarity on Realm's core use cases.
The Realm Function you mentioned has been deprecated and folded into the Event Handling capabilities of Realm PE and above.
Hope that helps and good luck with your app.