You do not get extra movement for rolling doubles while executing an event or community card, and you do not go to jail when rolling doubles while doing so.
Both interactions you describe apply to rolling doubles when rolling the dice to determine how far to move. While this isn't mentioned directly, it can be inferred from the Basic Gameplay paragraph (source) in the rules:
A player's turn consists of that player rolling the dice and moving their token clockwise that many spaces around the board. Then, the player undertakes an action corresponding to the type of space they landed on. Then, their turn ends and play passes to the next player clockwise from them.
If a player rolls doubles (the same value on both dice), then after they have resolved the action for the space they landed on they are allowed to roll the dice again and take an additional turn. If they roll doubles a second consecutive time, they are allowed to take a second additional turn. However, if the player rolls doubles a third consecutive time they are "caught speeding" – they must move their token directly to the "In Jail" section of the Jail space, and their turn is over.
The entire paragraph is about movement, and the rules further mention "being caught speeding" for the third toss, which reinforces the design idea behind the rule.
Then, there's a paragraph about executing a card's instructions, which allows for the interpretation that the instructions on a card are self-contained, and the game continues as normal once they're completed:
If the player lands on a Chance or Community Chest space, they must draw the top card from the appropriate deck. Then they follow the card's instructions, which will either involve paying or losing money or moving to a specific space. Once the instructions have been carried out, the card is returned to the bottom of the appropriate deck.
To specifically answer your two questions:
- You do not get to move an additional time by rolling doubles while executing the card. If you did, you would have to either re-use the result you just rolled to determine how much to pay, or re-roll the dice, both of which being highly unintuitive.
- You will not go to jail for rolling doubles, as you aren't "going too fast", since with the roll itself, you aren't going anywhere. Since you are never going to jail, there's no question whether you need to pay for the card's effect.
The rolling doubles rules are rather obviously installed to prevent any single player from moving an infinite amount of turns in a row while still allowing for players to land on a different amount of tiles in the same turn.