As already noted, the answer to all these questions is 'it depends'. There are many implementations, and they all differ in significant details.
One or more files. The organization of these files is part of the 'magic sauce' associated with a particular DBMS.
It varies a bit how the data is stored, but it is probably true to say that there are few or no DBMS that use one file per row for general working. The name management for a few billion (trillion, ...) records doesn't bear thinking about, all else apart.
This is again a key part of the 'magic sauce' for a particular DBMS. They all do it; there are many different ways to do it. SQLite uses radically different techniques from Informix, DB2, Oracle, Sybase, MS SQL Server, MySQL, PostgreSQL, Teradata, ... There are many techniques used; some use locking, others timestamp-based or multi-version concurrency control. There are different serialization properties for different DBMS.
I don't know of a DBMS that uses pure binary trees. Most DBMS provide multiple different organizations, but a very common one is a B+-tree (which is related to, but different from, a binary tree); another are various hash table organizations. One of the key objectives of most DBMS is to avoid having to rebuild things from scratch very often, and another is to avoid reading from disk as often as possible — by using caching and so on. Again, this is 'magic sauce' for a particular DBMS.