[SalesForce] Is it possible to assign Permission Sets Owned By Profile

You can get Object, Field, and SetupEntity permissions with the query outlined below. It gets you a lot of information...

static List<PermissionSet> getPermissions(Id userId)
{
    return [
        SELECT Name,
            (
                SELECT
                    SObjectType,
                    Field,
                    PermissionsRead,
                    PermissionsEdit
                FROM FieldPerms
            ),
            (
                SELECT
                    SObjectType,
                    PermissionsRead,
                    PermissionsCreate,
                    PermissionsEdit,
                    PermissionsDelete,
                    PermissionsViewAllRecords,
                    PermissionsModifyAllRecords
                FROM ObjectPerms
            ),
            (
                SELECT SetupEntityId, SetupEntityType
                FROM SetupEntityAccessItems
            )
        FROM PermissionSet WHERE Id IN (
            SELECT PermissionSetId
            FROM PermissionSetAssignment
            WHERE AssigneeId = :userId
        )
    ];
}

If you just care about CustomPermission, you could do something like:

static List<CustomPermission> getCustomPermissions(Id userId)
{
    Set<Id> assigned = new Map<Id, PermissionSet>([
        SELECT Id FROM PermissionSet
        WHERE Id IN (
            SELECT PermissionSetId
            FROM PermissionSetAssignment
            WHERE AssigneeId = :userId
        )
    ]).keySet();
    return [
        SELECT DeveloperName FROM CustomPermission
        WHERE Id IN (
            SELECT SetupEntityId
            FROM SetupEntityAccess
            WHERE SetupEntityType = 'CustomPermission'
            AND ParentId IN :assigned
        )
    ];
}

Or for all users:

static List<CustomPermission> getCustomPermissions()
{
    return [
        SELECT DeveloperName FROM CustomPermission
        WHERE Id IN (
            SELECT SetupEntityId
            FROM SetupEntityAccess
            WHERE SetupEntityType = 'CustomPermission'
        )
    ];
}

You are trying to update the existing GroupMember and PermissionSetAssignment records, rather than adding new ones. If you want to remove the old records, you will have to delete and replace, as they do not allow reparenting (based on your error message). You will also have to act on these related objects, not the User record itself. Something more like:

List<GroupMember> newGroupMembers = new List<GroupMember>();
for (GroupMember member : [
    SELECT GroupId FROM GroupMember WHERE UserOrGroupId = :mirrorUser
]){
    newGroupMembers.add(new GroupMember(
        GroupId=member.GroupId, UserOrGroupId=newUserId
    ));
}
insert newGroupMembers;

Same idea for PermissionSetAssignment:

List<PermissionSetAssignment> newAssignments = new List<PermissionSetAssignment>();
for (PermissionSetAssignment assignment : [
    SELECT PermissionSetId FROM PermissionSetAssignment WHERE AssigneeId = :mirrorUser
]){
    newAssignments.add(new PermissionSetAssignment(
        PermissionSetId=assignment.PermissionSetId, AssigneeId=newUserId
    ));
}
insert newAssignments;