The room bug should be fixed, if it wasn't fixed for the v.06a patch then it is for this coming version.
The hide button on mobile can be hard to press because of smaller screen size, all screens are gonna be a little different, but I can press it on my phone fairly easily.
We don't provide a Save option during scenes because we use RPG Maker and it does not automatically save things like event locations, sprite images, etc. So saving in the middle of an event and reloading a save can very easily break events as they won't be able to play properly upon reload. Most scenes, especially those with H-scene animations, are replayable through the replay room in MCs closet.
As long as you don't delete the game from your phone when you install the new version it will overwrite and update your current game and keep your saves. We cannot provide a save for mobile because of how the save files are stored, if/when we find a workable solution we will implement it