I'm at at loss trying to understand how normalizing it in each form ensures data integrity.
Data integrity is related to accuracy. If the data is accurate, we can trust it.
Separating lists of data enables us to more-accurately sort and filter data. (e.g. separating street address from suburb lets you filter for a particular suburb, whereas you couldn't do that before)
Including a primary key enables us to more-accurately retrieve the details of a record that may share a data value. (e.g. filtering LastName = "Smith" may retrieve multiple records, whereas filtering MemberID = "SMI0001" will retrieve the exact Smith we were looking for)
Separating tables removes redundant data, which means that the chances of updating anomalies appearing is reduced, making our data more-accurate (e.g. Member SMI0001 changes his address, if redundant data exists, the address needs to have many instances updated, whereas if there is no redundant data the address only needs to be updated once)
Moving calculated fields from the table into a query will change the calculation from being a manual calculation to an automated calculation - more-accurate (e.g. TotalFee would need to be calculated by the database operator as it's stored in the table - prone to human error. Whereas if it's calculated in a query, it can be automated using the Car Rate and Years Worked fields.)
In addition, you could also change the Years Worked field to Employment Date, which would be used to calculate Years Worked. This too would remove human error when updating, as when the data is stored in Years Worked it would require the Database Operator to change the Year for each record as the Driver completes another year of driving, a tedious task that would be prone to human error.
Also for a question like this, do we have to show each step of the normalization or just the final form?
Displaying your changes as a Data Structure Diagram would comprehensively show the transition from 1NF to 3NF in that you've shown the separating of fields, the primary keys & foreign keys, the relationships, the separating of tables and the removal of calculated fields.