Effects that modify or replace the cost to cast a spell (such as flashback or casting a spell without paying its mana cost due to suspend) don't affect the spell's mana value, so they won't change whether Lavinia's first ability restricts that spell from being cast.

For spells with {X} in their mana costs, use the value chosen for X to determine the spell's mana value.

If an effect allows a player to cast a spell without paying its mana cost, that player can't choose to cast it and pay its mana cost unless another rule or effect allows that player to cast it that way. However, if that spell also has additional costs that require mana, paying that mana will stop Lavinia's last ability from triggering.

Players may cast spells that they know Lavinia's second ability will counter. Any abilities that trigger when spells are cast will trigger and resolve if appropriate, and any effects that count spells cast will count those spells if appropriate.