Suppose a player in a Thursday Night Football game catches the eye of a team playing on the Sunday or Monday of the same Week, and a trade is quickly completed in the interim. The player is able to show up at his new team's game, would he be allowed to dress/play for that game? Is such a quick trade even permissible under league rules?
No. The commissioner's office is responsible for making sure teams are following protocol on trades. You can only play in one game per week.
Example - Percy Harvin playing for the Seahawks in a Thursday night game was not allowed to play for Jets that weekend.
Example - Will Allen played in 17 games in one year - he was traded to a team that already had their bye week when his team did not. I am not sure if he got paid more for it. https://www.pro-football-reference.com/players/A/AlleWi21/fantasy/2013/