Just saw this reply, the game saves your score as soon as the score screen pops up, and saves your settings as soon as you close the settings menu.
The downloadable build is out now! A Steam release would be cool obviously, but it's not possible (legally) since it's a fan-game, and the itch page works well enough. If you just want auto updates, I think you can use the itch app to do that (I haven't actually used it so not 100% sure). If you just want to be able to launch it from steam, you actually can do that, by clicking "Games" at the top, and then "Add a Non-Steam Game to my Library"