For an exact list of tables and columns Toad accesses when doing a Schema Compare, we do not have a list like that. Toad access the USER_ views, DBA_ views, and ALL_ views, however see further explanation below.
Toad specifies SYS.<view_name>, so you cannot create your own versions of these views for Toad to use instead.
To answer the question though, if you are comparing the schema that you are logged in as, Toad will use the USER_ views. If not, Toad will use the DBA_ views, if you have the SELECT privilege on them, and the ALL_ views if you don't. There are few views (such as DBA_SEGMENTS) which have no ALL_ views version.
In the schema compare, if you don't select an object type, then Toad will not select from the corresponding data dictionary view. For instance, if you don't select Outlines, Toad won't select from DBA_OUTLINES (this is another example of a data dictionary view that does not have an ALL_ version).
It also depends on what object types exist in your schema. For example, if you have "Tables" checked, but there are no tables in your schema, then we won't run any queries against USER\ALL\DBA_COLUMNS, USER\ALL\DBA_PARTITIONS, USER\ALL\DBA_LOBS, etc.
As a test on the SCOTT user (who does not have SELECT_CATALOG_ROLE) and ran a schema compare with every option checked. The only grant we had to make for it to work was:
Grant SELECT on SYS.ALL_AUDIT_POLICIES to SCOTT;
So this grant might probably work for you already. If you grant that one and still have errors, you can turn on Spool SQL or use SQL Tracker to see the queries that Toad is executing in the background, and it will also show the ones that has errors. The ones with errors would be the ones you would need to grant further access to for your user.
Spool SQL can be access via the Database Menu.
SQL Tracker can be access via the Database Menu | Monitor.
