The act states that the voting system should have a record that has capacity for auditing paper trails. Is also states it should produce permanent paper records that have the capacity for manual audits. That voters can have the opportunity to change or correct their ballot. So yes, as long as it is used properly it does mandate voter verified paper audit trails.
Yes, the HAVA does mandate that a paper audit trail is followed. The HAVA was a result of unscrupulous handing of the votes in regards to the presidency of President Bush. There was an inappropriate handling of the voting that caused the America Vote Act of 2002 to get implemented.