what happens if the data in bigquery changes, for example a row inserted? Are there any additional costs related to invalidation of the in-memory cache (ie the BI Engine) ?
Tables created in projects that have BI Engine enabled support streaming inserts, DML operations, etc. just like other BigQuery tables. Any queries you run using BigQuery itself incur regular query costs (https://cloud.google.com/bigquery/pricing#queries) but there are no extra charges beyond the storage costs (https://cloud.google.com/bi-engine/pricing) for BI Engine to manage the data for you behind the scenes.