Can I download this? I really like it so much.
One option is to download the game using the Itch App: https://itch.io/app
Note that you'll have to manually copy-paste your solutions from your browser into the downloaded version of the game. Eventually, I'd like to make it possible to share between browser and app, but there is at least one bug in the Itch App that prevents this from working with an Itch account.
It's not available as an .exe file. It opened in my browser.
Just so I understand, why were you looking for an EXE/stand-alone version of the game?
You're correct that there is no EXE; SIC-1 is a browser-based game for now. A desktop version could probably be built using Electron, but I don't have plans to do so. Launching from the Itch app sort of does this (it should open a new window that, while a web view, isn't just a normal browser window); it might even work offline (although all the leaderboards and whatnot won't work since they require Internet access).
Will my progress on the app be saved and will I be able to see my progress in my browser?
Sadly, no. The workaround is to just copy-paste your solutions.
I’m hoping to implement that eventually, but I haven’t done the work yet and there’s a bug in the Itch app that prevents it from working seamlessly anyway (although I have an active pull request with them on GitHub to fix that part).