If you play a solar eclipse card and draw a gherkin, do you get to keep the other card?

If the first card drawn is a Gherkin Attacks card, then the player's turn ends immediately and another card is not drawn. If however, a Gherkin Attacks card is drawn as the second card, then the first card can be kept (within the four card limit) and then the player's turn ends. If the first card can stop the Gherkin Attack, then this can be used before the player's turn ends.

if you play an Alliterating aliens card on the planet that you are on, do you lose a card as well?

Yes, all players at the planet lose a card, even if you were the one who played Alliterating Aliens.

if two players play cards outside of their turns at the same time, who gets to play?

The first player clockwise of the the person whose turn it is gets to play their card.

if you play a conjunction pair at the same time as another player, who gets to steal the turn?

If two players declare they are stealing another player's turn with a conjunction at the same time, then the first player clockwise of the the person whose turn it is gets to play their conjunction. This stolen turn could however then be stolen again by the other player with their conjunction.

if you play a wormhole, can the other player hold more than four cards once cards have been exchanged for a ship part?

Yes, they can hold the extra cards until their next turn. They can then either play them or discard them to end up with no more than four cards in their hand by the end of the turn.