If you know the offsets and sizes of the partitions that you used to have, then you can just recreate them in your new partition table. The filesystems are still where you left them, so once the partition table is reconstructed, everything should start working again.

If you don’t know that info, I am not sure what you do. Some kind of recovery utility could probably find your data without too much issue, but I am not too familiar with the current state of disk recovery tools.

That giveaway is also currently active on Steam as well, so one option is to grab the game on Steam and play it that way.

Sway is basically i3 but for Wayland, so guides for i3 may be somewhat helpful for you.

The man page is probably also worth a read: https://man.archlinux.org/man/sway.5

Short version: Sway/i3 doesn’t merely allow you to control your WM via hotkeys, it requires you to. Unless you know/have configured the hotkeys for opening a terminal or an application launcher, you won’t be able to do anything. As such, you are looking for a guide on how to configure Sway, rather than on how to use it. Once you know how to configure Sway, actually using it should be immediately obvious.