Best way to remap database login to new database user and keep previous database user
What is the best way to go about this? If database user dbuser02 is
  created and mapped to aduser01, what are the possible states dbuser01
  could be left in?


You can't create different database users mapped to the same login because a database user can be mapped to no more than one login.  The CREATE USER statement for dbuser02 failed so dbuser01 is still mapped to aduser01 and dbuser02 was not created.

I'm not exactly sure what you are trying to accomplish.  If your intent is to selectively grant database access to some, but not all, members of the mydomain\group01 and specify an individual default schema for each user, try the script below.

--add all members of mydomain\adgroup01 as role members
EXEC sp_addrolemember N'db_mydbrole', N'mydomain\adgroup01';
--create a default schema and user for desired mydomain\adgroup01 members
CREATE SCHEMA [aduser01] AUTHORIZATION [aduser01];
CREATE USER [mydomain\aduser01] WITH DEFAULT_SCHEMA=[aduser01];
CREATE SCHEMA [aduser02] AUTHORIZATION [aduser02];
CREATE USER [mydomain\aduser02] WITH DEFAULT_SCHEMA=[aduser02];