Star Trek: Deck Building - Upgrade Rules
Yes, you remove an upgraded card from your deck. No, you are not allowed to play a card immediately when you gain it. Normally, when gain a card it goes to your discard unless the card specifically tells you otherwise. No, if you upgrade a card from your bridge, the gained cards normally go to your discard area, unless the card says specifically otherwise.
From the Rulebook2.0 on Bandai's website, (page 24-25 Glossary):

Gain - Move a card from Starbase to your discard area (Some effects may move it to another area).
Upgrade - This is short for “Trash the designated card and gain a card from Starbase costing up to the designated amount more than the trashed card”. So you can trash a card of cost X and gain a card with a cost less then X.
Trash - Move a card to one of the following areas depending on where the card originated from:

If it is a starter card, remove it from the game.
If it is a Basic Character card, move it to its pile on Starbase.
If it is a card that started in the Starbase Deck, move it to the Starbase Discard Area.
If it is a Borg Card, move it to the Borg Card Pile