TPX profiles for static users are defined in TPXADMIN User Maintenance under option 5 - Maintain List of Profiles.
These profiles are used to assign the user options and build the TPX Menu from the session options for the userid when they sign on to TPX.
If none of the profiles defined to a static user are found in profile maintenance, will/can they be assigned a default profile?
No, there is no defined default profile for static users.
Note: This is different than dynamic users where a default profile can be specified within the SMRT System Features- Default dynamic user profile. This profile would be used if the user id not assigned any other profile during the signon process.