1) Mostly. But, you still need to set the fleet to Capture Planet or Capture Ships to get those sorts of ships in the fleet to behave properly. If the strategy is not Capture Planet, troop transports in the fleet will not drop their troops.
2) The only way to get troop transports to plunge into the fire is to send them to attack with _no_ warships. If any warships are present the transports will wait around for the WPs to be destroyed. If all warships die, the transports tend to plunge in.
3) It is not strange, it is how the AI works. If it did not stop firing once all WPs were gone, you would never be able to capture a planet. It does not tell all ships to "fire on planet". It issues orders to one ship at a time. So, once a ship destroys the Last WP, the planet is no longer a threat, and the troop transports can move right in. If there are no transports, then the default strategies tell ships to fire on unarmed planets, so they will proceed to glass it. Try playing around with firing priorities and targeting types in the Strategies.