1. Can "Slake the Thirst" Event card (Action: Exhaust your warlord to discard up to 3 cards at random from a target player's hand, then the player draws cards equal to the number of cards discarded) be used to target myself? Meaning can I use it on myself to exhaust my warlord and force my opponent to randomly chose 3 cards of mine to discard and draw 3 cards.

2. Sslyth Mercenary (Action: pay 2 resources to take control of this unit if it is at a planet. Any player may use this ability) - Is it possible to have bidding wars on this unit? Meaning one player pays 2 resources to take control of it only to have another player in the same action window do the same? Also if a player takes control of the unit after it is exhausted does it still remain exhausted?

1. Yes, since the event uses the word "player" (instead of, say, "opponent"), you can target yourself with it. Note that "discard at random from hand" does not necessarily mean you fan them out, face-down, and your opponent picks the cards - although that is usually treated as the easiest way to do it.

2. Yes. There is no limit on the number of times the ability can be triggered, so players can indeed effectively "fight" over control until one of them is out of resources by taking turns triggering the ability and taking control of it.

And yes; unless an effect specifically changes the ready/exhausted state of a card, it maintains its state.