1. Yes, the caster gets an Opportunity Attack
Since the creature is using its reaction to move away, that movement provokes an Opportunity Attack from an creature whose reach it leaves.
You do not provoke an opportunity attack when you teleport or when
something moves you without using your movement, action or reaction.
In case there was any doubt, Crawford has unofficially supported this ruling in a tweet.
If it leaves the caster's then they provoke from them as well and they can use a reaction to take the attack against them.
You can see Can I trigger an Opportunity Attack by forcing an opponent to move out of my reach? for more examples and discussion about forced movement provoking OAs.
2. No, you cannot use Blade flourish on the Opportunity Attack
Blade Flourish (XGtE, p. 15) only works when you take the Attack action:
Whenever you take the Attack action on your turn [...]
The bard did not take the Attack action this round, they took the Cast a Spell action instead. An opportunity attack is an attack, but it is made outside of an Attack action, so it will not trigger the Blade Flourish ability.
See What does upper-case-A-Attack action vs. lower-case-a-attack mean? for more information on the distinction between the two.