If I play a showdown card and play against the person next to me, what happens after the showdown? Does the person I challenged still get their turn, or was the showdown their turn?
I've never played this game, but I've now read the rules pdf from here
I couldn't find anything in the rules that indicated when the next player would start their turn! I can only assume passing a turn works in the same way as basic Uno rules, where play passes clockwise after a player plays a card.
The rules for showdowns say
The player who played the card takes the Showdown unit and places it between themselves and the next player in the current order of play (unless they played a Wild Showdown Card – more on that later).
the Wild Showdown Card is then expanded on later with
When you play this card, you get to choose which player to have a showdown with
There is nothing in these rules which indicates that taking part in a showdown is the target player's whole turn. It seems to me that a showdown just happens with the next player in turn order, who then takes their turn afterwards as normal.
If it was the next player's whole turn I would expect the rules for a wild showdown to suggest play continue from the player chosen for the wild showdown, but they do not.
As a general rule for games, if something is not stated clearly in the rules then it almost certainly is not a rule.