Alright here are some answers for your questions, I hope this helps.
1: No hotkeys for Trainer Card/Backpack/Pokedex implemented. You may create a suggestion for this under the Suggestions forum here: https://pokemonrevolution.net/forum/viewforum.php?f=7
2: No GTS in-game, however their is a Trade Forum
Red Trade Forum: https://pokemonrevolution.net/forum/viewforum.php?f=37
Blue Trade Forum: https://pokemonrevolution.net/forum/viewforum.php?f=109
3: If you click the wrench at the bottom right of your chat box, you can use that to temporarily leave any chats that you're currently in, including the All chat/Help chat.. etc
4: In-game you can find a small hand bag icon at the top right of your UI, click that. Once you do so, you can donate for coins in-game and from there you can use the coins to buy the MS Medallion which can be consumed. Once consumed you will have MS Status for 30 days.
Hopefully those answers helped, if you need anything else feel free to PM me on the the forums, or you can find me on Discord and in-game as well.